A monad (also called a triple), is a functor together with two natural transformations, satisfying certain conditions. See Wikipedia.
For basic notions, including modules over a monad, see Fresan’s survey of Durov’s work
http://mathoverflow.net/questions/55182/what-is-known-about-the-category-of-monads-on-set
http://mathoverflow.net/questions/19038/right-actions-of-operads-and-monads
nlab algebraic monad
http://ncatlab.org/nlab/show/comonad
http://ncatlab.org/nlab/show/monadic+adjunction
http://ncatlab.org/nlab/show/Eilenberg-Moore+category
http://nlab.mathforge.org/nlab/show/2-monad
nLab page on Monad