Category theory has been successful in mathematics and computer science. It is a very useful language to describe external properties of mathematical objects. It serves to compare mathematical theories when viewed as categories. Eventually category theory is also nice and has its own concepts: adjunctions, yoneda Lemma, Kan extensions..
Enriched category theory (ECT) intends to described richer interactions of objects. Roughly speaking, interactions between objects a,b of a (classical) category are captured by the set, Hom(a,b) of arrows from a to b. In an enriched category the Hom(a,b) may not be a set anymore but roughly any other good kind of object. A theory of enriched categories has been developed (Eilenberg, Kelly, Dubuc, Street, Lawvere...). It is similar to the classical theory of categories. The applications of ECT are multiple: metric spaces, automata, domains, sheaves, commutative algebra, and much more.
I will recall quickly the basis of categories, introduce enriched ones, and discuss applications.