I understand monads

@jay cred given. now explain them!

Oh, man, called out.

OK. This isn't going to be perfectly precise because it's too abstract for me to spend the time explaining in all of its gory detail, besides, my category theory is weak. What follows is very hand-wavey.

There are two groups (G,*) and (H,+). There is a function f: G -> H such that for all x, y in G f(x * y) = f(x) + f(y). f is a homomorphic -- structure preserving -- function.

Now we're going to abstract the notion of a homomorphic function over groups to a function M that preserves some form of structure over 2 mathematical objects -- sets, groups, topological spaces, what-have-you). M is a morphic 'function' (not that M has to be a function or the objects that it acts on have to be sets).

F is a functor (I don't think I've used capital F yet) -- a special case of a morphism. F is analogous to f in that given two objects, J and K, and given all morphisms('functions') T: J -> K, F preserves identity morphisms and the composition of morphisms.

Still with me?

There are two functors, F' and F'' such that (F'x,y) = (x,F''y). F' and F'' are adjoint. F' is left adjoint to F''. A monad is the composition F'' o F'.

Bet you thought I was going to talk about Haskell, eh?

Oh, neat. I just went and looked up functors and monads in the Haskell docs and my comment explains exactly what they do.

Functors preserve identity and composition:
fmap id == id
fmap (f . g) == fmap f . fmap g

Monads are the composition F'' o F':
return a >>= k == k a

didn't understand a word of it.

Mmmm. From a programming perspective they're probably good for...storing state or building big operations out of little functions, I'd guess.

They're so abstract it's hard to relate them to much of anything.