By Grumblefloof and RocketLord
First things first: The most important Mutants and Masterminds rules for you to have in mind at any given point in time are 1) Power Level Caps, 2) Trade-offs, 3) Extra Effort, and 4) Hero Points.
Most everything else is just "Roll a D20, add a bonus, try to hit a target number (Difficulty Class, or DC)."
This page does not exist to replace reading the first 8 chapters in the Mutants and Masterminds 2E core book, but to help understanding these rules.
Power Level Caps
Power Level Caps and Trade-offs are covered in Chapter 1 of the core book, the red chapter, pages 24-26.
Power Level is how Mutants and Masterminds achieves combat balance, and to a lesser extent, overall character balance.
Most of your traits (but not all of them) are capped at a certain value relative to your Power Level. Power Level is usually abbreviated to PL, followed by a number. E.g., PL10 means Power Level 10, which means the character caps out at Power Level 10 caps.
On Freedom City Play By Post, you start with two PL10 slots and one PL7 slot. It is recommended that your first character be one of the PL10s slots, since it is easier to build a character with PL10 caps and 150PP available, than a character with PL7 caps and 105PP available.
It is also highly recommended that you build your character in such a way as to actually meet your PL caps, so your PC isn't frustratingly ineffectual in play. If your character does not meet their offensive PL caps, you will have a hard time hitting or damaging enemies, and if you do not meet your defensive PL caps, it will be much easier to hit or hurt you than it should be.
In most other RPGs, you start with a foundation, and you build on it, as high as you can stack the bonuses.
Mutants and Masterminds isn't like that. You start with a ceiling, your Power Level, and you build up to that ceiling.
Since your PC is PL10, you want your best possible Attack Bonus, Save Difficulty, Defense Bonus, and Toughness Saving Throw Bonus (what Mutants and Masterminds has instead of Hit Points) to all reach +10.
There are several ways to do this. None of them are "better" than the others, but some of them are cheaper, because they're more limited in scope and usefulness.
You can be a generalist, or you can specialize. Specialization is cheaper, but does not apply to all situations.
Attack Bonus
Your Attack Bonus determines how well you hit a target. To hit try to hit someone, you roll a d20 and add your Attack Bonus to the roll to determine if you hit someone or not.
Examples
You could buy your base Attack Bonus that gets used for all attacks you make up to the full +10 cap for 20PP. Then you have +10 Attack no matter what attack you're using - punching a guy in the face, shooting him with a gun, whacking him with a baseball bat, picking up a dumpster and throwing it at your target, whatever: You will add +10 to your Attack Roll to hit your target.
Or you could decide to specialize. Buy Base Attack up to +5 for 10PP, 3 ranks of the Attack Focus [Melee] feat for another 3PP, and one rank of the Attack Specialization [Swords] feat for 1PP.
Then you have a +10 attack bonus when swinging a sword at a guy, or a +8 bonus when using some other melee attack, or a +5 bonus when using any kind of ranged attack.
Save Difficulty
Damage and the effect your powers have on their target depends on what you use to attack and sets the DC a target must beat on a Toughness or Exotic Save Roll to avoid being affected. The Save Difficulty of a given attack sets the full DC that target must reach on their save: A Toughness Save to avoid taking Damage is 15 + Save Difficulty = Toughness Save DC, while an Exotic Sav to avoid being affected is 10 + Save Difficulty = Exotic Save DC.
A melee attack will use your Strength Bonus as the Save Difficulty, while other DC Bonuses will depend on the ranks of your Powers. The total Save Difficulty is capped by your PL, so for a PL10 character, it cannot go above +10.
Examples
A generalist might buy their Strength ability score up to 30, costing 20PP, giving them a +10 Save Difficulty to all attacks made using their Strength, whether its when hitting someone with a punch, with a sword, throwing a car at them or whatever else.
The specialist could buy their Strength ability score up to 18, costing 8PP, then buy the Damage power at Rank 6 with the Mighty feat, costing 7PP, that lets you add your Strength bonus to the Save Difficulty of the power, for a total +10 Save Difficulty, for a total cost 15PP. While cheaper, you will only have the full +10 Save Difficulty when you use the given power, not with all attacks. This Damage power could for instance be used to show you using a sword, claws coming out of your hands, coating your hands in a strange energy, or even just a martial arts move.
Another specialist could buy a power at rank 10, for instance Blast, costing 20PP, which would give you a Save Difficulty at +10, but only for the attacks you make using the given power. This might cost more than the specialist above, but you can use the attack at a range. This Blast could be eye beams, which you could use in a situation where your hands are bound, or something entirely else.
Defense Bonus
Your Defense Bonus determines how difficult you are to hit. In order to hit you, an attacker's Attack Roll (1d20+Attack Bonus) must be equal to or higher than your Defense Bonus + 10. Half of your Defense Bonus is called the Dodge Bonus, and it will only be applied if you're aware of an attack. In other words, if you're surprised by an attack, you will be easier to hit.
Examples
You can buy your Base Defense Bonus up to +10, costing you 20PP. In that case, you will have a total Defense of 20 in most cases, or a Defense of 15 if you're surprised.
Alternatively, you could buy Base Defense Bonus +4, costing you 8PP, and 6 ranks of the Dodge Focus feat for 6PP, costing you a total of 14PP, and giving you a total Defense Bonus of +10. The downside is that only half the bonus from the Base Defense Bonus applies when you're surprised, not the bonus from the Dodge Focus feat, so when surprised, you would only have a Defense of 12.
Toughness Saving Throw Bonus
Your Toughness Saving Throw Bonus determines how difficult you are to hurt. If an attacker has hit you, you must roll a Toughness Save Roll to avoid being hurt. In order to avoid being hurt, your roll of 1d20 + Toughness Bonus must be equal to or greater than the attack's Save Difficulty + 15. Toughness is a bit different in that you can't buy a base Toughness bonus, like with the Base Defense and Attack Bonuses. Your Constitution modifier is added to your Toughness bonus, which can also be improved by buying the Protection power (or any of its variations) or the Dodge Roll feat.
Examples
A generalist could buy their Constitution ability score up to 30, costing 20PP, giving them Toughness +10. The Constitution modifier gives a lot of other benefits, and in this way, they would always have the full Toughness bonus.
But perhaps you don't want your character to have super human constitution, it just doesn't make sense for them. Then, you could buy your Constitution ability score up to 16, costing 6PP, giving you a +3 to your Toughness Saving Throw Bonus. On top of that, you could buy 5 ranks of the Protection power for another 5PP to give you another +5 to your Toughness Saving Throw Bonus, and 1 rank of the Dodge Roll feat for 1PP to get another +2, for a total of +10 to your Toughness Saving Throw Bonus, at a cost of 12PP. This would give you the same total bonus, but maybe the Protection power comes from an armor, so it won't apply if you're not wearing the armor, and the Dodgle Roll feat's bonus doesn't apply if you're surprise attacked. It is cheaper, but more specialized. Note that we have special house rules for the Dodge Roll feat found here.
Exotic Saving Throws
The Fortitude, Reflex and Will Saving Throws are generally known as the Exotic Saving Throws when grouped together. As previously mentioned, the DC of an Exotic Save is equal 10 + Save Difficulty, with the Save Difficulty usually being the rank of the power that affects you. Each of the 3 saves receive a bonus from an Ability Score (Constitution for Fortitude, Dexterity for Reflex and Wisdom for Will), and additional bonuses to each save can otherwise be purchased at a cost of 1PP per +1.
Unlike the Toughness Save and Defense, the Exotic Saves do not work by trade-offs, with their maximum bonus (Ability Score + additional save bonus purchased) being equal to your PL+5. A PL10 character can, for example, have Exotic Saves at no more than than a +15 bonus to your Save Roll.
Rules As Written, you could potentially increase all of your saves to this bonus, if you have the necessary PP, but unlike your Attack Bonus, Save Difficulty, Defense Bonus and Toughness Saving Throws, this is most likely not something you want. It becomes very difficult for a GM to affect your character in any way if all of your character's Exotic Saving Throws are maxed, which, while awesome for you not getting hurt or stopped at all, also means that you are unlikely to find many GMs interested in involving your character in threads, since action scenes will usually either offer your character no challenge at all or become too difficult for any of the other involved characters to handle.
As such, the general suggestion is that your exotic Saving Throws either average your PL-2, or, for particularly hardy individuals, your PL. It is a rule of thumb, of course, and a Paragon might have a particularly high Fortitude Save, or a skilled Acrobat might have a high Reflex Save, but keeping your saves within this average range gives you options for standing out in certain saves, while not needing the GM to annihilate everyone else to still keep you challenged.
Examples
A PL10 generalist has all their Exotic Saves at +8.
A PL10 acrobat could have Fortitude +6, Reflex +11, Will +7, averaging a +8 bonus.
A PL10 paragon with maximized Constitution (for a total +15 bonus) could have Fortitude +15, Reflex +5, Will +7, averaging a +9 bonus. Higher than the suggested PL-2, sure, but staying below a +10 average for the PL.
In most other games, Sword Guy is just as good as The Generalist in overall combat, but he's even better than The Generalist with a sword.
In Mutants and Masterminds, The Generalist pays a premium to be good with everything, including swords, while Sword Guy gets a discount for only reaching his maximum potential with a sword, and not being as good as The Generalist at non-sword attacks.
You don't have to hit your PL caps under all circumstances, just under your best possible circumstances - Using your preferred attack, against your preferred enemy, in your preferred environment, etc.
Specialization is cheaper than being a generalist because your "best possible circumstances" come up less often.
The exact list of everything affected by your Power Level Caps are found in the core book, on page 24
Trade-offs
Power Level Caps and Trade-offs are covered in Chapter 1 of the core book, the red chapter, pages 24-26.
Since your PC is PL10, you want your best possible Attack Bonus, Save Difficulty, Defense Bonus, and Toughness Saving Throw Bonus to all reach +10.
That was a lie, or at least a half-truth, to ease you in.
Technically, your Attack Bonus is on an axis with your Save Difficulty, and your Defense Bonus is on an axis with your Toughness Saving Throw Bonus, and each of those pairs has to average to 10 (or lower, but you don't want to go lower).
So you can have Attack +10 and Save Difficulty +10, or Attack +15 and Damage 5, or Attack +5 and Save Difficulty +15, or anything in between. You don't even have to stick to the same trade-off for different attacks, so maybe you have an accurate but weak attack with Attack +15 and Save Difficulty +5, but then another that's more balanced, at Attack Bonus +12 and Save Difficulty +8.
Trade-offs are a way to 1) Keep every character from looking the same, and 2) More accurately represent certain characters and archetypes.
Note that, as per our house rules, at least 1/3rd of your total Defense or Attack Bonus must be bought with Base Defense or Attack Bonus. This doesn't mean that you have to buy at least 4 points of Base Attack Bonus if your character is PL10. Rather, it means that the total bonus for your attack with your highest Attack Bonus must have at least 1/3rd of its bonus based on Base Attack Bonus. So, if you have an Attack with Attack Bonus +15 and Save Difficulty 5, and another with Attack Bonus +7 and Save Difficulty +13, then your Base Attack Bonus must be at least 5, 1/3rd of the highest attack bonus.
Examples
All as PL10.
You might want to create an acrobatic hero that is difficult to hit. In that case, you would want to increase your Defense Bonus, at the cost of lowering your Toughness Saving Throw Bonus. Let's say you go for a +4 Defense Bonus / -4 Toughness Saving Throw Bonus trade-off. Then, your total Defense Bonus could reach +14 and your total Toughness Saving Throw Bonus could reach +6. At least 1/3rd of your total Defense Bonus must be bought from Base Defense Bonus, so you buy Base Defense Bonus +5, for 10PP, and then buy 9 ranks of the Dodge Focus feat for another 9PP. For a total cost of 19PP, your Defense Bonus is now +14. You buy Constitution 18, for 8PP, to get +4 to your Toughness Saving Throw Bonus and 1 rank of the Dodge Roll feat for 1PP, giving your another +2 to your save, for a total Toughness Saving Throw +6 for 9PP. Having part of your Toughness Saving Throw Bonus coming from Dodge Roll even further goes to show that you're an acrobatic hero that rolls with the punches you do take.
You want someone that's a great shot, which will in turn meet your attack is weaker. You decide to get a +2 Attack Bonus / -2 Save Difficulty trade-off for one of your attacks. You need at least 1/3rd of your Attack Bonus to come from your Base Attack Bonus, and you don't care about hitting in melee, so you buy Base Attack Bonus +4 for 8PP, then buy 8 ranks of the Attack Focus [Ranged] feat for 8PP, giving you a total Attack Bonus with ranged attacks of +12 for 16PP. Then, you buy a Blast power at rank 8 for another 16PP, which will have a Save Difficulty of +8.
Extra Effort
Extra Effort and Hero Points are covered in Chapter 6 of the core book, the pink chapter, pages 120-122.
With Extra Effort, you take a free action (so you can only do it on your own turn, but it doesn't cost any actions), get something special, and then suffer a level of Fatigue on the next round.
There's a number of possible uses of Extra Effort, but the most common use is a Power Stunt: You temporarily gain a single Power Feat on one of your powers of your choice. The most common choice for that power feat is Alternate Power, which allows you temporarily do something you can't normally do.
The temporary power feat sticks around for the entire scene, unless you grabbed a new Alternate Power, in which case it goes away sooner if you swap to the original power you gave the Alternate Power feat, or another power in the array, in case you added another Alternate Power to an existing array.
These Alternate Powers gained through Power Stunting represents things that you can always do, but its not something that you use often enough for it to make sense to spend PP on. Maybe its a special weapon in your battlesuit that you only need to pull out once in a while, maybe its a creative use of your powers or something that takes some effort actually do. A wizard might have 100s of spells, but if there's only 4 or 5 that see any regular use, then you have those as your regular attacks in an array, and for the rest, you stunt. In short: Just because its not on your character sheet, it doesn't mean you can't do it.
Examples
You are facing a group of weaker enemies and you have a Snare power, which allow you to trap one enemy. You want to trap them all with just one attack. In order to try to trap all enemies at once, you use Extra Effort to Power Stunt, adding an Alternate Power with the Area Extra to the hit them all. Since the original power and the Alternate Power have the same PP cost total, the new power will be weaker, but since its weak enemies, it doesn't matter.
You have an array with a number of electric powers. You want to shut down a big super computer, but its shrugging off all damaging attacks you throw at it! You then use Extra Effort to Power Stunt an Alternate Power to the Array, choosing the Nullify power, applying it to electronics in order to shut down the computer.
Facing an overwhelming foe, you need an attack that can take him down in the next hit. Your battlesuit has a bunch of weapons, but none of them are strong enough to really hurt him, even if he's easy to hit. You use Extra Effort to Power Stunt a new power off your Weapons Array, a secret gun that has +15 Save Difficulty / +5 Attack Bonus.
Hero Points
Extra Effort and Hero Points are covered in Chapter 6 of the core book, the pink chapter, pages 120-122.
Hero Points is a special resource that only Player Characters have access to. NPCs, no matter if they're enemies controlled by the GM or sidekicks controlled by the player, do not have them. As a standard, you enter a thread with 1 Hero Point and can earn more by performing heroic actions or by triggering your complications. There is no limit to the amount of Hero Points you can save, but they are not carried between different threads. The number of Hero Points you have in one therad does not in any way affect the number of Hero Points you have in another. You can start a thread with additional Hero Points by buying ranks in the Luck feat, with each rank letting you start the thread with one extra Hero Point. Luck cannot go higher than 1/3rd of your Power Level under normal circumstances.
You can spend Hero Points for a number of different effects, see page 121 of the core book for the full list. Spending a Hero Point is a Reaction, so you can do it at any time, even during another player's turn or during a NPC's turn.
One of the many things you can do with a Hero Point is "Heroic Feat", where you gain a single Feat (normal or power) for 1 round, in essence the same effect as Power Stunting for Extra Effort, but also extending to regular Feats. In this way, you could suddenly gain the use of a feat that might be not normally be useful, but be incredibly useful in just this situation.
Another thing you can do is immediately recover from a level of Fatigue. Because you can do this, a lot of people will short-hand spending a Hero Point for a Power Stunt.
The big takeaway here is that "what your character can do" is not limited to just what's written on their sheet.
You don't have to buy a massive power Array with a dozen Alternate Power feats to cover every conceivable use of your character's powers.
You don't have to buy every single feat you think your character might have use for at some point.
You can just buy the stuff you'll use most often, buy a few Luck feat ranks to start each game with some extra Hero Points, and then use Extra Effort and Hero Points to temporarily grab stuff during play as you need it. And if you find yourself temporarily grabbing the same thing over and over again, then maybe you spend some earned Power Points on buying it permanently.