A Maybe monad just says the value may be Something(x) or Nothing. You won't know until you run the computation. If you use flatmap and give a function that takes an x and gives a Maybe[x], the monad will first map into Maybe[Maybe[x]] and then flatten into Maybe[x]. The computation has not happened until you execute it and internally all the functions have been composed together.
A List[A] just says, give me a function A->List[B], and I will flatmap (flatten `compose` map) it. So it maps each element into a possibly empty list of Bs, and then flattens it by concatenating them.
You can define your own monads, and as long as they obey the laws of monads, you get a bunch of stuff for free.