A feat is a type of advantage (a positive and distinctive trait) that is used in games descended from Dungeons & Dragons 3rd edition, including most D20 System games and Pathfinder. Characters in d20 games sometimes have the option to gain new feats when they level up.
The word feat is defined as an impressive act or accomplishment, and many feats in D&D 3rd edition did grant access to new types of action and ability (such as the Power Attack feat, a new type of attack that let players reduce their accuracy to increase their damage). However, not all feats necessarily provided new actions, skills, or powers, and instead moderately changed the way that characters could interact with the standard rules of the game in certain circumstances. Feats in more recent games like Dungeons & Dragons 5th edition are more commonly in this latter category.
Feats generally stand alone, but may have certain traits (including other feats) or statistic scores as prerequisites. Some specific feats may be able to be taken multiple times, and in those cases the feat rules will usually describe what the impact of doing so will be.
Feats add character rules on top of the rules provided by a character's race and class. Feats may interact with each other or other rules in interesting ways that game designers may not expect, which makes them useful for powergaming.