These are longer games that exist mostly to tell a story in which you play a role as the lead character. They are "Role Playing Games" for your console or computer!
There is typically a character creation system, a combat system, and some sort of level system involved, however, the exact form these things take will vary. Turn based RPGs are very common, but real-time RPGs are becoming more popular in recent years. Believe it or not, there are both clean and Adult oriented RPGs, so be careful when clicking around if you're under 18!