In an RPG the player takes the part of a person living in the game world. Goals are much less clear-cut than in an FPS, but all the same there are goals. As you conquer obstacles and complete quests, or missions, your abilities improve and your character "grows".
Classic examples of RPGs include:
This is not to be confused with Role Playing which is, simply put, assuming the Role of your character as much as possible the way an actor would become the part while in front of the camera.