In algebraic geometry, modules over commutative rings can be seen as "bundles" on the spectrum of the ring. In detail, if $R$ is a commutative ring and $M$ is an $R$-module, then there is an associated sheaf of modules $\widetilde{M}$ on the scheme $\mathrm{Spec}(R)$ whose module of global sections is exactly $M$. The stalk of $\widetilde{M}$ at some point $\mathfrak{p} \in \mathrm{Spec}(R)$ is the localization $M_\mathfrak{p}$, the fiber is given by $M_\mathfrak{p}/\mathfrak{p} M_\mathfrak{p}=M \otimes_R Q(R/\mathfrak{p})$, which is a $Q(R/\mathfrak{p})$-vector space. Although not completely correct, one may imagine a (finitely generated) $R$-module $M$ as a bunch of these vector spaces. (It is more accurate to consider the stalks $M_\mathfrak{p}$.)
For example, consider the $\mathbb{Z}$-module $\mathbb{Z}/6$. The associated sheaf has support $\{(2),(3)\}$, and the fibers are $\mathbb{Z}/2$ at $(2)$ and $\mathbb{Z}/3$ and $(3)$. Thus, this module becomes free of rank $1$ when restricted to its support. Now consider the $\mathbb{Z}$-module $\mathbb{Z} \oplus \mathbb{Z}/5$. The support of the associated sheaf is the whole spectrum $\mathrm{Spec}(\mathbb{Z})$. The fiber at $(5)$ is $\mathbb{Z}/5 \oplus \mathbb{Z}/5$, a $2$-dimensional vector space over $\mathbb{Z}/5$. The fiber at $(0)$ (usually called the "generic fiber") is the $1$-dimensional vector space $\mathbb{Q}$ over $\mathbb{Q}$. If $p \neq 5$, the fiber at $(p)$ is the $1$-dimensional vector space $\mathbb{Z}/p$ over $\mathbb{Z}/p$. You may imagine that the $\mathbb{Z}$-module $\mathbb{Z} \oplus \mathbb{Z}/5$ is "made up" out of these vector spaces.
However, in order to understand a module, it is often not enough to look at its fibers. There are many subtle properties and invariants of modules; for example flatness, torsion, semisimplicity, projectivity, invertibility, local freeness, depth, projective dimension, injective dimension, etc.
Here is one of the uncountable reasons why it is interesting to look at modules: If $R,S$ are commutative rings such that the categories of modules are equivalent, $\mathsf{Mod}(R) \simeq \mathsf{Mod}(S)$, then $R$ and $S$ are isomorphic. In fact, $R$ can be constructed from $\mathsf{Mod}(R)$ as the center of this category. It follows that, at least theoretically, we can understand a commutative ring by looking at its category of modules. In fact, ring theory benefits a lot from module theory. There is a theorem (due to Serre) which states that a commutative local ring is regular if and only if its global dimension is finite, i.e. the set of projective dimensions of modules is bounded. This implies that that localizations of regular local rings are regular again, which is a purely ring-theoretic fact but which is hard to prove without the theory of modules.
The theory of modules over non-commutative rings is closely connected to representation theory, which is about the "symmetry in linear spaces". For example, if $G$ is a group and $k$ is a field, then we may construct the group ring $k[G]$, and the category of left $k[G]$-modules is isomorphic to the category of $k$-linear representations of $G$. Large parts of representation theory are actually module theory.
