A feat is a special ability earned by a player character through advancement. Unlike class features that every character in a class gets (e.g. all paladins gain detect evil as a spell-like ability), a player can choose any feat for which the character qualifies.
In Dungeons & Dragons 3rd Edition, which popularized the concept, characters earn feats at first level and at every third level. Bonus feats outside the regular advancement scheme can come through class features or racial abilities; a human fighter will get a bonus feat for being human, another for being a first-level fighter, and then another every second level he takes in fighter. Bonus feats are usually restricted to certain prerequisites, such as the fighter's bonus feats, which must be selected from a combat-focused list.
Feats can be found in other games, where they may have another name. Regardless of system or name, feats allow players to further personalize their characters.