One of the purposes of category theory, as far as I have understood, is to provide a way to talk about mathematical structures independently of their points or elements: For instance, we can describe groups, rings and other structures not by the points they contain but how the maps in and out of them behave.
But a category in itself can also be considered an algebraic structure, like groups and rings: A category also has points, only two kinds of points: Objects and morphisms. So in a sense, the language of category theory itself is highly dependent on referring to the points of the categories in question. For instance, the functor property
$$F(g \circ f) = F(g) \circ F(f)$$
for all $f \in \text{Hom}(x,y)$, $g \in \text{Hom}(y,z)$, is certainly dependent on points. So the idea is, just as categories let us describe algebraic structures without mentioning their points, can we describe categories themselves without mentioning their points?
To do this, let $C$ be some small category. Then the collection of all objects in $C$, call it $\text{Ob}$, forms a set. Likewise, the collection of all morphisms in $C$, call it $\text{Mor}$, also forms a set. We have two evident set maps
$$s,t : \text{Mor} \longrightarrow \text{Ob}$$
The map $s$ sends a morphism $f: x \to y$ to its source $x$, and the map $t$ sends $f$ to its target $y$.
We also have a set map
$$\text{id} : \text{Ob} \longrightarrow \text{Mor}$$
sending an object $x$ to the identity morphism on $x$ which we denote by $\text{id}(x)$ (we reserve the notation $\text{id}_X$ for the identity function on the set $X$)
A little more complicated map is the composition map
$$ * : \{ (g, f) \in \text{Mor} \times \text{Mor} : t(f) = s(g) \} \longrightarrow \text{Mor}$$
sending a pair of morphisms $(g,f)$ with $s(g) = t(f)$ to the composition $g * f$ (We reserve the symbol $\circ$ for composition of set functions). The domain of the map $*$ looks a little complicated at first sight, but we recognize it as just the pullback of the diagram
$$\begin{matrix} & & \text{Mor} \\ & & \downarrow \small t \normalsize \\ \text{Mor} & \underset{s}{\rightarrow} & \text{Ob} \end{matrix}$$
i.e. we have a pullback square
$$\begin{matrix} \text{Mor} \times_{\text{Ob}} \text{Mor} & \overset{p_2}{\longrightarrow} & \text{Mor} \\ \small p_1 \normalsize \downarrow & & \downarrow \small t \normalsize \\ \text{Mor} & \underset{s}{\rightarrow} & \text{Ob} \end{matrix}$$
Thus $*$ is nothing but a map
$$ *: \text{Mor} \times_{\text{Ob}} \text{Mor} \longrightarrow \text{Mor}$$
To summarize: We have seen that the data of a (small) category can neatly be described as a pair of sets $\text{Ob}$ and $\text{Mor}$ together with four maps $s: \text{Mor} \to \text{Ob}$, $t: \text{Mor} \to \text{Ob}$, $* : \text{Mor} \times_{\text{Ob}} \text{Mor}$ and $\text{id} : \text{Ob} \to \text{Mor}$.
We need some axioms however. First we need to control the source and target of composite and identity maps. If $s(f) = x$, $t(f) = y$, $s(g) = y$ and $t(g) = z$, we want $s(g \circ f) = x$and $t(g \circ f) = z$. This can be summarized by requiring that the following diagrams commute:
$$ \begin{matrix} \text{Mor}\times_{\text{Ob}} \text{Mor} & \overset{*}{\longrightarrow} & \text{Mor} \\ \small p_2 \normalsize \downarrow && \downarrow \small s \normalsize \\ \text{Mor} & \underset{s}{\longrightarrow} & \text{Ob} \end{matrix} $$
$$ \begin{matrix} \text{Mor}\times_{\text{Ob}} \text{Mor} & \overset{*}{\longrightarrow} & \text{Mor} \\ \small p_1 \normalsize \downarrow && \downarrow \small t \normalsize \\ \text{Mor} & \underset{t}{\longrightarrow} & \text{Ob} \end{matrix} $$
We also want $s(\text{id}(x)) = x$ and $t(\text{id}(x)) = x$, which can be translated into the following commutative diagram:
$$ \begin{matrix} \text{Ob} & \overset{\text{id}}{\longrightarrow} & \text{Mor} \\ \small \text{id} \normalsize \downarrow & \overset{\text{id}_{\text{Ob}}}{\searrow} & \downarrow \small t \normalsize \\ \text{Mor} & \underset{s}{\longrightarrow} & \text{Ob} \end{matrix} $$
Here $\text{id}_{\text{Ob}}$ is the identity map on the set $\text{Ob}$, not to be confused with the map $\text{id} : \text{Ob} \to \text{Mor}$.
Now that we have control of the source and target of composites and identities, we want to state the actual category axioms, namely associativity of composition and behavior of identities with respect to composition. I leave it to you to check that this is equivalent to requiring that the following two diagrams commute:
$$ \begin{matrix} \text{Mor} \times_{\text{Ob}} \text{Mor} \times_{\text{Ob}} \text{Mor} & \overset{\circ \times_{\text{Ob}} \text{id}_{\text{Mor}}}{\longrightarrow} & \text{Mor} \times_{\text{Ob}}\text{Mor} \\ \small \text{id}_{\text{Mor}} \times_{\text{Ob}} \circ \normalsize \downarrow & & \downarrow \small \circ \normalsize \\ \text{Mor} \times_{\text{Ob}} \text{Mor} & \underset{\circ}{\longrightarrow} & \text{Mor} \end{matrix} $$
$$ \begin{matrix} \text{Mor} & \overset{(\text{id}_\text{Mor}, \text{id} \circ s)}{\longrightarrow} & \text{Mor} \times_\text{Ob} \text{Mor} \\ \small (\text{id} \circ t, \text{id}_\text{Mor}) \normalsize \downarrow & \overset{\text{id}_\text{Mor}}{\searrow} & \downarrow \small * \normalsize \\ \text{Mor} \times_\text{Ob} \text{Mor} & \underset{*}{\longrightarrow} & \text{Mor} \end{matrix} $$
So, what we have shown is that we can define a small category $C$ to be the data of two sets $\text{Ob}$, $\text{Mor}$ and four maps $s,t,*,\text{id}$ (with the domains and codomains mentioned before) such that all of the diagrams above commute. Thus we have defined a small category without mentioning any of its elements!
But by looking at categories in this way, we discover something much more exciting: $\text{Ob}$ and $\text{Mor}$ are just sets, i.e. objects in the category $\mathbf{Set}$, and likewise, $s,t,*,\text{id}$ are just functions, i.e. morphisms in $\mathbf{Set}$. If we just replace the category $\mathbf{Set}$ with any ambient category $A$ having enough pullbacks, we will get a completely new notion of a category! This is what we call a category internal to $A$, namely the data of two objects $\text{Ob}$, $\text{Mor}$ in $A$ and four morphisms $s,t : \text{Mor} \to \text{Ob}$, $* : \text{Mor} \times_\text{Ob} \text{Mor} \to \text{Mor}$, $\text{id}: \text{Ob} \to \text{Mor}$ in $A$ such that all the diagrams above commute. What we have done in category theory before is studying the boring special case of a category internal to $\mathbf{Set}$.
Now we can do all sorts of crazy things, like considering a category internal to the category of groups or even the category of small categories. I encourage you to try to unwrap these concepts!
Just as we have developed ordinary category theory before, i.e. defined functors, natural transformations, limits, adjoints and so on, we can try to develop ordinary category internal to some fixed category $A$. In particular, how should we define a functor $F: C \to D$, where $C$ and $D$ are categories internal to $A$? It is not too hard to convince oneself that this should be the data of two morphisms $F_0 : \text{Ob}_C \to \text{Ob}_D$ and $F_1 : \text{Mor}_C \to \text{Mor}_D$ such that some diagrams commute. For instance, the requirement that the two diagrams
$$
\begin{matrix} \text{Mor}_C & \overset{s_C}{\longrightarrow} & \text{Ob}_C \\ \small F_1 \normalsize \downarrow & & \downarrow \small F_0 \normalsize \\ \text{Mor}_D & \underset{s_D}{\longrightarrow} & \text{Ob}_D \end{matrix}
$$
$$
\begin{matrix} \text{Mor}_C & \overset{t_C}{\longrightarrow} & \text{Ob}_C \\ \small F_1 \normalsize \downarrow & & \downarrow \small F_0 \normalsize \\ \text{Mor}_D & \underset{t_D}{\longrightarrow} & \text{Ob}_D \end{matrix}
$$
commute, generalize the requirement that if $f: x \to y$ is a morphism in some ordinary category $C$ and $F: C \to D$ is a functor between ordinary categories, then $F(f)$ should be a morphism from $F(x)$ to $F(y)$.
I won't write out the rest of the functor diagrams, but I encourage you to do it! Another thing to think about is how to define a natural transformation in the internal sense. An interesting question is the following: Do the collection of categories and functors internal to $A$ form a category?
I think the concept of internal categories illustrates how easy it is to abstract stuff using category theory. Another generalization of categories is enriched category theory, where the hom-sets are not sets but objects in any (sufficiently nice) category. The possibilities are endless.
