I'm a master's student in algebra and am trying to collect a list of concrete situations where rings (and hence, modules) are natural objects of study, or useful, as a way to motivate them. Not all of these are situations where rings are strictly necessary. I've seen more intrinsic reasons for studying rings elsewhere on this site and am not looking for that. Here rings don't have to be unital nor commutative. Here is my current list:
Algebraic number theory: Some equations of number-theoretic interest, such as Pell's, can be factored, e.g. $x^2 - n y^2 = (x + y \sqrt{n})(x - y \sqrt{n})$ in $\mathbb{Z}[\sqrt{n}]$. Hence, in this case, one way to understand this equation is by understanding the ring $\mathbb{Z}[\sqrt{n}]$.
Field theory: A lot of constructions done for fields are passed through rings. For instance, $\mathbb{Q}(\sqrt{2})$ can be constructed as the quotient ring $\mathbb{Q}[x]/(x^2 - 2)$. I believe this can be attribued to the importance of polynomials in field theory, and the fact that these form rings and not fields. Another example is that (polynomial) rings are used in Artin's proof of the existence of algebraic closures.
Representation theory of groups: There is an isomorphism between the categories of representations of groups $\rho: G \to \text{GL}_n(V)$ over a field $k$, and modules over the group algebra $k[G]$. (Also that of $G$-modules). Hence, if one is knowledgeable about modules over rings, then one suddenly knows a lot about representations of groups: Their category is abelian, one can take tensor products, and so on. Further example: One can then see that Maschke's theorem for a field $k$ and group $G$ says when the group algebra $k[G]$ is semisimple. But the semisimple rings are those with global dimension zero (and we otherwise know some things about them by Wedderburn-Artin), hence (with the help of rings) we see that Maschke becomes a statement on the global dimension of the category og $G$-representations over $k$.
Group (co)homology: The invariant subgroup functor is isomorphic to the functor $\text{Hom}_{\mathbb{Z}[G]}(\mathbb{Z}, -)$. Hence, we get isomorphisms $H^n(G, M) \simeq \text{Ext}^n_{\mathbb{Z}[G]}(\mathbb{Z}, M)$ and similarly for $H_n(G, M)$. Thus, group (co)homology can be reduced to module theory over $\mathbb{Z}[G]$.
Algebraic geometry: One way to understand a variety is by looking at its coordinate ring (or its structure sheaf, which contains many rings). For instance, one way to study the circle $x^2 + y^2 - 1$ is by studying its coordinate ring. Furthermore, studying the variety at a point can be done by looking at its stalk. For instance, the variety is smooth at a point iff the corresponding ring of germs is a regular local ring.
Linear algebra: Studying a linear function $T: V \to V$ of a vector space $V$ and its invariant subspaces is the same as studying $V$ as a module over the polynomial ring $k[x]$, where the action of $x$ is that of $T$. Hence, this kind of linear algebra is the same as module theory over $k[x]$, and by understanding this ring, one may better understand this kind of linear algebra. This idea is used together with the fact that $k[x]$ is PID, and the structure theorem of fin.gen. modules over PIDs to prove matrix decompositions like the Rational Canonical Form and the Jordan Canonical Form.
Analysis: Many spaces of functions, like $C^\infty(M)$ and $C_c^\infty(M)$ for $M$ a smooth manifold, its rings of germs $C_p^\infty(M)$ at points $p \in M$, different kinds of spaces of continuous functions $C(X)$, $C_0(X)$ and $C_c(X)$ for $X$ a (LCH) topological space all turn out to be rings, and thus this might be the right way of viewing them. Also, C*-algebras are rings, so one can avoid (some) duplication of work by having some knowledge of rings before studying C*-algebras.
Filter theory: For a (nonempty) index set $I$ and a field $k$, there is a strictly increasing bijection between the filters on $I$ and the ideals of the product ring $k^I = \prod_{i \in I} k$. Hence, one may transfer (which?) knowledge of rings to study filters. Furthermore, since ultrafilters correspond to maximal ideals, one may characterize the ultrafilter lemma (in ZF) as being a Krull-like statement of rings of this kind. Furthermore, it gives a more pedagogical construction of ultrapowers of fields. Constructing an ultrapower ${^*}k = k_\natural$ of a field $k$ over an index set $I$ is the same as taking a quotient $k^I/\mathfrak{m}$, where $\mathfrak{m}$ is a maximal ideal containing the direct sum ideal $\bigoplus_{i \in I} k$. (There is a generalization of this to ultraproducts of rings, but that defeats the point of motivating them.)
Homological algebra: By the Freyd-Mitchell embedding theorem, some results of abelian categories can be reduced to showing them for module categories.
(some kind of) Category theory: If $\mathcal{C}$ is a preadditive category, then for any object $A \in \mathcal{C}$, $\text{End}_\mathcal{C}(A)$ becomes a unital ring. Some information of the object can be read from its endomorphism ring. For instance, if the endomorphism ring is local, then the object must be indecomposable. Furthermore one gets a Krull-Remak-Schmidt-theorem in general if one assumes every object in the decomposition in question has a local endomorphism ring. (At some point here one assumes more of $\mathcal{C}$ than just being (pre)additive.)
Do you know of any other examples?
