Specifically, given some algebraic object $$A$$, a representation of $$A$$ is a vector space $$V$$ and a a structure-preserving map $$\rho\colon A \to \mathrm{End}(V)$$, the ring of endomorphisms of $$V$$. That is, we're embedding $$A$$ inside a ring of matrices to "linearize" $$A$$. The usual first example that students see of representation theory is in the case of finite groups, where $$A$$ is a finite group and $$\rho$$ is a group homomorphism.
Studying representations of a ring can also be thought of as studying modules over that ring. For a ring $$R$$ and a vector space $$M$$, a representation $$\rho \colon R \to \mathrm{End}(M)\colon r\mapsto \rho_r$$ gives $$M$$ the structure of a left $$R$$-module, with action $$r\cdot m = \rho_r(m)$$.