Avatar progression and advancement is typically termed experience or leveling. This is most often seen in role-playing games, but it's a mechanic that is quickly spreading to many other genres. It's seen most recently crossing into the first-person shooter genre, in titles such as Dead Island and Borderlands.
In it's most basic form, progression takes place by "leveling up" after a certain set of criteria are met. Typically this is defeating enough enemies to raise a quantity, typically termed experience points. Leveling up usually gives benefits such as increasing attributes or awarding skill points.
Avatar progression has an obvious motive (the player's) and an ulterior motive (the designer's). Simply put, if your avatar progresses in power, it makes it easier to defeat enemies, complete challenges, and advance through the game. This is typically balanced by the designer by placing progressively harder enemies in the path. The designer's motive (conscious or not) for this mechanic is that as the avatar gains power, the player has invested time in it, and is less likely to want to give up that investment. This can effectively lock in a player, especially in MMOs, where a player may not want to give up his max-level character.
Level Balance - Spreading difficulty, allowing player to choose, or designed difficulty by spawning enemies of the appropriate level