Monadicity theorems like Beck's give sufficient conditions on a functor $U \colon B \to A$ with a left adjoint F to be equivalent (in the slice category Cat/A) to the forgetful functor out of the category $A^{U F}$ of algebras for the monad $U F$.

Aside from their obvious usefulness in universal algebra, one important application is in descent theory: suppose you have a category C and a C-indexed category E (i.e. a pseudofunctor $E \colon C^{\mathrm{op}} \to \mathrm{Cat}$) such that

1. Each $f^* = Ef$ has a right adjoint $f_!$, and

2. The Beck--Chevalley condition holds: E takes any pullback square in C to an isomorphism in Cat whose mate is again an isomorphism.

Then (this is due to Bénabou and Roubaud) for a morphism f in C, the category of descent data for f is equivalent to the category of algebras for the monad $f_!f^*$. In particular, f is of effective descent if $f_!$ is monadic. See the nLab page on monadic descent for details.

I believe the original application had C the category of commutative rings and $E \colon R \mapsto R\mathrm{-Mod}$.