Actually, I understand these types in terms of containers that override standard method of function application for their contents. In fact, there may be no contents, or several items, or nothing at all, or something strange, but the container behaves like there is a value of some type inside it. In these terms: Maybe - container that may or may not contain something [a] - container that contains several values and applies function to all of them State - container that has some other "secondary" value in it. IO - container that remembers passed functions and later will ask user for value, then apply functions to it.
* Monad: A monad M allows a function of n arguments (for n >= 0) to be applied to n M values; in addition, if the function returns an M value itself, you can combine that result with the arguments in a sensible way.
Here, what does "sensible" mean? What do we override? Also, would it be right to say that Arrow is a container for functions that overrides function chaining?