From the Wikipedia:
"The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures.
Instead of focusing merely on the individual objects (e.g. groups) possessing a given structure, category theory emphasizes the morphisms — the structure-preserving mappings — between these objects. It turns out that by studying these morphisms, we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the group homomorphisms. A group homomorphism between two groups "preserves the group structure" in a precise sense — it is a "process" taking one group to another, in a way that carries along information about the structure of the first group into the second group. The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms." (http://en.wikipedia.org/wiki/Category_theory)