A monad on a category is an endofunctor equipped with two natural transformations satisfying associativity and the unit laws . The two transformations are a unit and a multiplication. Equivalently a monad is a monoid in the endofunctor category, which is the organising slogan for compositional effects. The weaker applicative-functor sits between a plain functor and a monad, whereas a relative-monad relaxes the endofunctor to a functor . Its algebras are maps compatible with , and these form the Eilenberg–Moore category, whereas the free algebras form the Kleisli category. Every adjunction induces a monad , and every monad arises this way. The prototype is the distribution monad, also called the Giry monad, whose Kleisli maps are the channels of a markov-category. Other standard examples are powerset for nondeterminism, for partiality, list, and state, each choosing what “impure” means.
References
Saunders Mac Lane, Categories for the Working Mathematician (1971), ch. VI (monads, Eilenberg-Moore, Kleisli)
Michèle Giry, “A categorical approach to probability theory”, in Categorical Aspects of Topology and Analysis, LNM 915 (1982)