Let $j\colon A \to C$ be a functor between categories. Its codensity monad is the right Kan extension $Ran_j j$ of $j$ along itself, if this exists (as it certainly does when $A$ is small and $C$ is complete).
The name comes because $j$ is codense just when its codensity monad is the identity. Thus, in general, the codensity monad “measures the failure of $j$ to be codense”.