"Dungeons & Dragons" is the most popularly known role playing game (a.k.a. roleplaying game or role-playing game). People use the acronym RPG most of the time. While most RPGs center around elaborate combat rules, at its heart it is really improvisational acting. You are acting the role of a character thrust into various situations and must always be answering the critical question "what do you do?"
RPGs run the spectrum from the traditional "dungeon crawl" / "kill people and take their stuff" of "Dungeons & Dragons", to the almost entirely free form and diceless "Amber", and to the "Storyteller" series ("Vampire", "Mage", "Werewolf", etc.) sitting somewhere in the middle. World of Warcraft is firmly in the "kill people and take their stuff" camp.
The major advantage of computerized RPGs is that they handle all the rules, mechanics, and die rolling for you, letting you focus on just the role playing / whacking monsters on the head. The disadvantage is that the computer is not nearly as flexible as a human Game Master, and thus computerized RPGs tend to be shallow and linear with little actual "role playing". Online RPGs (such as WoW) partially fix this problem by at least giving you other humans to interact with, but the Game Master is still a computer, and thus quests and NPC interactions are shallow.