You're in a mind. A dream, but it's not yours. You delve deep inside and traverse bizarre alien geometry, fighting off the projections of whoever's subconscious you're in. You open safes to look at their deepest hidden memories, and get to the heart of their deep-seated fear and anger to solve their problems, and in the process, fundamentally change the dreamer themselves.
No, Psychonauts isn't Inception: The Game, released far before we were talking about dreams and cool stunts. It's much more inventive, much more imaginative, and much, much more fun.