Help me gain some intuition for the simplex category I've been reading about the simplex category on nLab and Wikipedia, and I have to say, I'm extremely confused. Maybe it's just because I'm fairly new to category theory and I don't have the requisite knowledge of homology/cohomology to know what this is actually used for, but I'm struggling to get any intuitive sense of how these objects are used.
Wikipedia defines the simplicity category $\Delta$ to have the collection of all non-empty finite ordinals as its objects, and weakly order-preserving maps as its morphisms. The augmented simplicity category $\Delta_a$ is defined to have all finite ordinal as its objects, and order preserving maps as its morphisms. nLab gives a bunch of other, fairly high level definitions that I won't reproduce here. My question is though: what does any of this have to do with simplices? I just can't see what the connection is. Beyond that, why are these categories useful to us in the way that they are in algebra and topology?
 A: 
My question is though: what does any of this have to do with simplices?

Write $[n]$ for the finite ordinal $\{ 0 \le 1 \le 2 \le \dots \le n \}$, considered as an object of the simplex category $\Delta$. At the most basic level the idea here is to think of this object as an $n$-(dimensional) simplex, and in particular to think of the points $0, 1, \dots n$ as its vertices, with the ordering giving the simplex an orientation. Order-preserving maps $[n] \to [m]$ then correspond to "degenerate subsimplices." Looking at how this works for small $n$ should help make this clearer:

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*$[0] = \{ 0 \}$ is a point / vertex / $0$-simplex.

*$[1] = \{ 0 \le 1 \}$ is a directed edge, or a $1$-simplex. There are two "face" maps $[0] \to [1]$ corresponding to the two vertices of the edge, and a "degenerate" map $[1] \to [0]$ which collapses an edge down to a vertex. In fact if we take the full subcategory of the simplex category on $[0]$ and $[1]$ and consider presheaves on that we get a slight variation of the category of directed graphs in which every vertex has an "identity edge." So, as a rough first pass, the category of simplicial sets is like a category of "higher-dimensional directed graphs."

*$[2] = \{ 0 \le 1 \le 2 \}$ is an oriented triangle, or a $2$-simplex. There are three "face" maps $[0] \to [2]$ corresponding to the three vertices of the triangle. There are also three injective "face" maps $[1] \to [2]$ corresponding to the three edges (with compatible orientations), and three non-injective "degenerate" maps which collapse the edge to one of the three vertices.

You may wonder why we need these "degenerate" maps; thinking just in terms of the geometry of simplices it may make sense to restrict attention to strictly order-preserving (equivalently, injective) maps only. This can be done and gives rise to the notion of a semisimplicial set (or more generally semisimplicial object). I am not the person to ask in detail about why simplicial objects turn out to be a better thing to study.
In any case here's a rough first pass describing what a simplicial set (a presheaf on the simplex category) is: in general the category of presheaves $\widehat{C} = [C^{op}, \text{Set}]$ on a (small) category is the free cocompletion of $C$, meaning that it is obtained from $C$ by "freely adjoining colimits." So the category $\widehat{\Delta}$ of presheaves on the simplex category consists of objects obtained by "freely gluing together simplices." This gives some idea of what simplicial sets are supposed to look like as spaces (namely, they look a bit like simplicial complexes), and can be formalized using the geometric realization functor $\widehat{\Delta} \to \text{Top}$, which loosely speaking turns formal colimits of simplices into actual colimits of simplices in $\text{Top}$.

Beyond that, why are these categories useful to us in the way that they are in algebra and topology?

This would require a much longer answer and I'm not really the one to write it; a full answer would really require a textbook. Loosely speaking, simplicial objects turn out to be a "nonlinear" generalization of chain complexes, and can be used to write down "resolutions" of nonlinear objects such as spaces (e.g. Cech nerves) in a way that generalizes how chain complexes can be used to write down resolutions of linear objects such as modules. But this won't be a helpful answer to you until you have some experience in homological algebra (which would be a useful prerequisite for thinking about simplicial stuff).
The simplest really generally useful application of this idea is that the free abelian group on a simplicial set can be used to write down a chain complex (via the Dold-Kan correspondence); this is how singular homology is traditionally defined, using the singular simplicial set $\text{Sing}(X)$ of a topological space. $\text{Sing}$ turns out to have a left adjoint given by geometric realization, and this was one of the motivating examples of adjoint functors when Daniel Kan introduced them in 1958.
