Informally, it is a general way of describing algebraic structures.

A monad is a Category $\mathcal{C}$, together with a Functor $T: \mathcal{C} \to \mathcal{C}$ from that category to itself, equipped with two Natural transformations $\eta: 1_\mathcal{C} \Rightarrow T$ and $\mu: T^2 \Rightarrow T$ satsifying unit and associativity axioms.

An important construction is an Algebra over a monad

See here

Examples

• Monad for monoids where $T$ maps sets to their Kleene star, and its algebra is a Monoid
• Monad for small categories

Free functor

Distributive laws

https://www.youtube.com/watch?v=mw4IhOLhDwY

The 2-category of monads inside any 2-category, with a Monad functor

Every adjunction gives rise to a monad (Adjunction) and every monad gives an adjunction between the underlying set of the monad, and the category of algebras over that monad. Also every monad gives an adjunction to the Kleisli category.

video that explains Kleisli category . Next vid (good intro to understand monads too)