Role-playing games, or RPGs, tend to belong to one of three categories. For more information, see the Wikipedia.
Emphasize real-time combat. Characters have stats and skills which are improved by gaining experience through fighting or progressing through the story. Generally feature lots of useable weapons and other randomized loot. Common settings include medieval, futuristic, and cyberpunk.
Japanese RPGs (JRPGs)
Almost always feature turn-based combat. Graphics are in the Anime style, and the protagonists are usually teenagers out to save the world from an ancient evil. Their settings almost always either feudal Japan or super-high-tech future/space.
These games are about playing a character in a story - combat may not even be present. They generally have branching storylines, multiple endings, and lots of choices with later consequences. This is the least-common form of RPG.