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A module over a commutative ring R is the natural generalization of a vector space over a field: an abelian group M equipped with a scalar multiplication by elements of R satisfying the same axioms as vector spaces. Unlike vector spaces, modules over general rings need not have bases, and the absence of division in R creates fundamentally new phenomena — torsion elements, non-free modules, and the failure of dimension theory.
If you know linear algebra, you know vector spaces: abelian groups where you can multiply elements by scalars from a field. A module over a commutative ring R replaces the field with an arbitrary ring. The axioms are identical — distributivity, associativity of scalar multiplication, and 1·m = m — but the consequences are profoundly different because elements of a ring need not have multiplicative inverses.
The most illuminating example is that ℤ-modules are exactly abelian groups. Given any abelian group G, the map n·g (adding g to itself n times) defines a ℤ-module structure, and this is the only one possible. So every theorem about modules over ℤ — the structure theorem, the theory of torsion, the classification of finitely generated modules — is simultaneously a theorem about abelian groups. The Fundamental Theorem of Finitely Generated Abelian Groups is precisely the structure theorem for finitely generated ℤ-modules.
The key new phenomenon in modules is torsion: an element m ∈ M is torsion if rm = 0 for some nonzero r ∈ R. In a vector space, this forces m = 0 (multiply both sides by r⁻¹), but over a general ring, r need not be invertible. The ℤ-module ℤ/nℤ consists entirely of torsion elements. A module with no nonzero torsion elements is called torsion-free. Free modules (those with a basis) are torsion-free, but the converse fails in general — though it holds over PIDs, where torsion-free and finitely generated implies free.
Modules are the natural language of commutative algebra because ideals themselves are modules. An ideal I ⊆ R is a submodule of R (viewed as a module over itself), and the quotient ring R/I is a quotient module. Module homomorphisms generalize both ring homomorphisms and linear maps. The entire machinery of exact sequences, tensor products, and homological algebra operates at the module level, and virtually every construction in commutative algebra — localization, completion, primary decomposition — is formulated in terms of modules rather than just rings.