In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. They are a Semigroup with identity.
They are also a one-object Category, with arrows corresponding to elements in the monoid, and composition corresponding to monoid multiplication. Video
There is also the category of all monoids, with arrows corresponding to Monoid homomorphism (defined in the natural way).
It can also be constructed as an Algebra over a monad