Fine, I will be the only person not trolling in this thread.
THE definition:
Role Playing Games is a genre of video games that feature some combination of the following gameplay elements:
a. Character development - Player controls one or more in-game characters, who obtain new abilities or improve their characteristics over the course of the game.
b. Combat - Player controlled character(s) engage(s) in combat with other in-game entities.
c. Exploration - Player controlled character(s) can traverse the game world and find new locations, characters, and objects.
d. Dialogue - Player controlled character(s) can engage in dialogue with other in-game characters.
Element a is required, b, c and d are usually present but vary in their importance.
The term originates from pre-computer Pen and Paper game sessions and is a misnomer for the video game genre. Role Playing Games within the context of Pen and Paper sessions meant that every participant in the session takes on a particular "role" (e.g. dwarf paladin, elf rogue) which is separate from their actual persona. For PnP sessions, this made a lot of sense, as enjoyment was obtained from the "acting out" of the roles and the interaction between them. Had PnP players stuck with their own persona every time, both of those advantages would be lost.
Once computers and consoles came along, one of the earliest types of games became games that featured the 4 elements above, in various combinations. Because of certain connections to the PnP scene (same audience, fantasy settings, character development, dungeon crawling, etc), they became known as RPGs. However, in the video game context, this was not an accurate name anymore. Early video game RPGs (Ultimas, Wizardries, etc) did not support actual "role playing" to any significant degree. Moreover, with video game RPGs, "role playing" was no longer an essential requirement for enjoyment, as it was with PnP sessions. Since each video game RPG offered its own virtual world, story, and gameplay, the player could play each one from his own personal perspective (and not that of some character), and still enjoy it, which would be impossible with PnP sessions.
Because of this, the vast majority of games released as RPGs do not actually have true role-playing. Therefore, the name should not be interpreted literally, but is simply a label for games that fit the beginning of this definition.