I'd say Shiver did a great job of providing an explanation. Beginner we all know. Intermediate riders should know how to ride any well-trained horse without giving it conflicting cues, and so on. Intermediates also know how to tack, feed, and otherwise care for horses. Advanced riders can ride even horses that need a lot of training yet and they don't get nervous or worry about falling off. They also should know how to train these horses.
I'll never call myself advanced, but I have gone to describing myself as intermediate in the last year or so. Frankly, I'd be happy always being a beginner, and there is so much to learn about horses that I really do feel that way much of the time, but I probably am intermediate now.
Life is hard enough on its own, please be kind.