Question

Higher Categories and monoids

Answer 1

An enriched category is an ordinary category with extra structure on the hom-sets between objects! For example if for all objects a and b in a category C the hom-set Hom(a,b) has the structure of a group, ring, simplicial set, field or even topological space then your category C is enriched over (given structure). It turn out that categories enriched over topological spaces or simplicial set are good models for higher category theory. Neither is perfect, but both have their uses.

One of the difficult parts of higher category theory to get stuck on is learning the straightening-unstraithening theorems by Lurie, where you switch between different models for the hom-sets, since they're weakly equivalent but not isomorphic.

Monoidal categories is a different topic. If a category has a tensor product satisfying the monoid axioms you have a monoidal category, for example topological spaces and the smash product. If you're working with a monoidal category an individual object can have a monoid structure! For example studying monoids in the category of abelian groups is studying rings! This also generalizes to higher categories but that is a bit more technical and less intuitive.

Answer 2

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)

Answer 3

The three concepts ( higher cats, monoidal cats, enriched cats ) all play together in some fashion.

As explained in the other comments, a monoidal category is a category equiped with a 'tensor product' structure, like the cartesian product for sets, tensor product for vector spaces or modules in general, product topology for topological spaces, etc, etc.

The idea is to give an axiomatization of what happens in these cases and study these categories as an object in themselves.

Enriched categories over a monoidal category are just categories in which the hom sets are replaced by objects in a monoidal category. So now the homs can be something different than sets, the existence of the tensor product of the monoidal category is what allows you to form a 'composition' of morphisms, even if now there might not be proper elements in the homs. Or maybe there are elements but now we also have more structure, like a topology which allows you to think about the shape of the space of the hom sets and study those in itself. This is an important point of view in higher cats.

So higher cats are sometimes modeled as certain kinds of enriched categories depending on what you want to do with them. On the one hand the easiest most immediate thing you can do when you want to do 2-cats is think about categories enriched in categories, these will give you a strict 2-cat.

For higher categories some people ask the categories to have some sort of topology  so that you can now talk about the homotopy of these spaces. You can have the mental image that 1-categories are categories where the morphisms form graphs ( just nodes and edges ) and the idea of equivalence in category theory essentially dawns to understanding and comparing these graphs ( objects are unimportant ! ).  So now the idea is that homs form spaces and you want to compare then now these spaces with each other in the same way.

To go back to monoidal categories, you can also construct higher categories directly. Take a monoidal category M, now consider the category C with one object. The morphisms of this object with itself is the category M. For any two 'morphisms' m,n in M, there is a composition mxn ! Additionally, there are morphisms between morphisms, the morphisms between m and n in M! and these are nicely compatible with the 'composition' given by the monoidal product.

So you can actually go back fully and think about monoidal categories as their generalizations as coming from certain higher categories.

All these topics are studied on their own and in interaction with each other depending on the questions you want to answer. I really think the nlab is an awful source if you are just getting into the stuff , for pedagogical and formative reasons, I would stick with Baez and other more accessible sources to get a good idea of what is going on.