Let's start with monoidal categories. For many categories we're interested in, there's a functorial way of combining two objects to get a third one which also satisfies some other properties. Some key examples are the tensor product of abelian groups, and Cartesian products of set, topological spaces, and categories in their respective categories. A category with this structure we call monoidal, because the objects roughly form a monoid. We call this the monoidal product. (Note that there can be different monoidal structures on the same category, but we'll only be interested in one at a time. And that even if a category admits categorical products, that may not be the monoidal product we're interested in as in the case of the tensor product of abelian groups.) Instead of unitality and associativity holding exactly, they only hold up to coherent natural isomorphisms. For instance, for sets: (X x Y) x Z != X x (Y x Z) , but they are naturally isomorphic, and this isomorphism interacts nicely with the analogous one for unitality: X x {\*} \~= X \~ = {\*} x X.

In a regular (locally small) category C, for any two objects X,Y, there's a set of homomorphisms C(X,Y), and we can think of composition as a function (of sets) C(X,Y) x C(Y,Z) -> C(X,Z). And this satisfies a number of naturality and coherence conditions which basically encode the axioms of a category. In many other categories, we get something similar but where the hom sets can be viewed as objects of some other category. If A is a preadditive category (eg an abelian category) then A(X,Y) is not only a set, but has the structure of an abelian group, and composition is bilinear. So it can be viewed as an abelian group homomorphism A(X,Y) \\otimes A(Y,Z) -> A(X,Z). And this satisfies the same naturality and coherence conditions. Another example is topological spaces - given spaces X, Y, the set of continuous maps Top(X,Y) has the structure of a topological space using the compact-open topology. And composition gives a continuous maps Top(X,Y) x Top(Y,Z) -> Top(X,Z) satisfying the same conditions.

In general, given a monoidal category V with monoidal product \\otimes, we say a category C is enriched in (V, \\otimes) if the morphisms C(X,Y) is an object of V, and composition can be encoded as a morphism in V: C(X,Y) \\otimes C(Y,Z) -> C(X,Z). And these satisfy the usual conditions. So a locally small category is exactly the same as a category enriched in sets and the Cartesian product.

(I need to go, but I'll add more later in a reply)