We consider actions of finite monoids on left topological compact semigroups by continuous endomorphisms. Our study of this situation is inspired by and has implications for Ramsey theory. We prove that for a class of monoids strictly extending the class of R-trivial monoids such actions always embed a finite configuration defined from the monoid only. This result, through ultrafilter methods, implies Ramsey theoretic statements; for example, it gives a new proof of the Furstenberg--Katznelson Ramsey theorem. It also makes it possible to define a notion of Ramsey monoid and give a characterization of those in terms of algebraic properties of the monoid. This characterization implies a negative answer to a question of Lupini on possible extensions of Gowers' Ramsey theorem.