Why is it called the category of representations? Let $A$ be a (Hopf) algebra. Let $C_A$ be a category whose objects are $A$-modules and whose morphisms are $A$-linear maps.
This category is called "the category of representations".
My question is: why is it called the category of representations? Is there any relation to other area of mathematics? What does it represent? Is it called representations only for "Hopf" algebras?
 A: In representation theory people often use the word representation as a synonym for module, and for good reason.  A representation of a Lie algebra is the same thing as a module for its universal enveloping algebra, a representation of a quiver is the same as a module over its path algebra, etc.
One important example is that a representation of a finite group is the same thing as a module over its group algebra, which is naturally a (cocommutative) Hopf algebra. Perhaps that can motivate the terminology.
A: For a given group $G$ and field $F$, the category of group representations of $G$ (in the sense of group homomorphisms from $G$ into $GL(n,F)$) is equivalent to the category of finite dimensional modules over the group ring $F[G]$.
Similarly, the category of $F$-algebra representations of an algebra $A$ (in the sense of ring homomorphisms from $A$ into $M_n(F)$) is equivalent to the category of finite dimensional $A$ modules.
These are the two main connections between "representations" in the sense of "modeling an object as a set of matrices" and related categories of modules.
