Generalizing Monads

I'm sure this is all well-known by the hardcore category theorists, but I don't have any references because I just invented it myself. This is going to involve some higher-order category theory, but I'm not going to be theoretical about it so don't worry if you have trouble grasping basic (first-order) category theory. I'll give some suggestive sounding theory, but I haven't verified it or bothered to figure out what other people call these things.

( So what *is* a monad? )