Dummit & Foote makes the following definition:
Let $R$ be a commutative ring with identity. An $\mathbf R$-algebra is a ring $A$ with identity together with a ring homomorphism $f:R\to A$ mapping $1_R$ to $1_A$ such that the subring $f(R)$ of $A$ is contained in the center of $A$.
I understand the definition, and see some examples (any ring with identity is a $\mathbb Z$-algebra, the polynomial ring $R[x]$ is an $R$-algebra, etc.) but I am not really seeing the motivation behind this definition. Why are we interested in this object? To contrast, I can see modules over rings as a generalization of vector spaces over fields (the former of which I am not too familiar with, the latter of which I am fairly familiar with). But this definition just seems a bit artificial and I am not comfortable with it.
Also, I assume there are other types of "algebras," since there is made a distinction here that this is an $R$-algebra. Is it the case that there is a general structure called an "algebra", and this is a specific type of an algebra? I do remember from my operator theory class that there were Banach and $C^*$-algebras - are these also "algebras" in the sense of the algebraic structure?