By definition, an $R$-algebra is a ring homomorphism $f: R \to S$. For example, if $R=\mathbb Z$ and $S= \mathbb Z / n \mathbb Z$ then the projection $k \mapsto k \mod n$ is a ring homomorphism so that $\mathbb Z / n \mathbb Z$ is a $\mathbb Z$-algebra. I think the point of an algebra is that it's a bit like a module in that we extend its structure by adding a ring that is acting on it. In the case of modules, we start with an abelian group and in the case of algebras we start with a ring.
Now for my question: I've been trying to come up with a non-finitely generated $R$-algebra but couldn't. Can someone help me and give me an example? Thank you.