The category Cat of small categories consists of all small categories as objects, and, functors between small categories as morphisms. The composition of morphisms in Cat is the functor composition, and, associated with each small category, the identity functor acts as the identity morphism. Now, Cat is indeed a category, since $\hom(\mathcal{C},\mathcal{D})$ the class of all functors from $\mathcal{C}$ to $\mathcal{D}$ is a set. The proof of this fact can be found here.

Here are some of the basic properties of Cat:

1. It has arbitrary products
2. It has arbitrary coproducts
3. Initial object exists: the initial object is the empty category and the associated empty functor.
4. Terminal object exists: the terminal object is any trivial category and the associated constant functor into the trival category.
5. It has pullbacks. See this entry. So it has equalizers, and therefore, it is complete.
6. however, it does have coequalizers. This, together with 2 above, shows that it is cocomplete.

Remarks.

• If we replace functors in $\hom(\mathcal{C},\mathcal{D})$ by natural transformations between pairs of functors from $\mathcal{C}$ to $\mathcal{D}$ and composition of morphisms the horizontal composition $\circ$ of natural transformations, then we again end up with a category (provided that both $\mathcal{C}$ and $\mathcal{D}$ are small). Indeed, every natural transformation $\eta$ between two functors from $\mathcal{C}$ to $\mathcal{D}$ is a set function from the set of objects of $\mathcal{C}$ to the set of morphisms of $\mathcal{D}$ As a result, $\hom(\mathcal{C},\mathcal{D})$ is a subcollection of the set of all functions from $\operatorname{Ob}(\mathcal{C})$ to $\operatorname{Mor}(\mathcal{D})$ and hence a set. For more detail, please see this entry.
• In fact, Cat has the structure of a 2-category, where the small categories are the $0$ cells, the functors between them are the $1$ cells, and the natural transformations between parallel functors are the $2$ cells.
• If we remove the requirement that each object in Cat be small, then $\hom(\mathcal{C},\mathcal{D})$ may no longer be a set, and we end up with a large category.