Modules are abelian groups with an added notion of multiplication by elements in a ring. They generalize abelian groups, which are modules over the integers, and vector spaces, which are modules over fields.
Rigorously, a left module is defined as an abelian group $M$ paired with a ring $R$ with a binary operation from $\cdot:R\times M\rightarrow M$ satisfying the following axioms for all $m,n\in M$ and $r,s\in R$:
$r\cdot(m+n)=r\cdot m+r\cdot n$
$(r+s)\cdot m=r\cdot m+s\cdot m$
$(rs)\cdot m=r\cdot(s\cdot m)$
A right module is defined similarly by rewriting the axioms with the ring elements on the right side.
Modules often arise in the study of commutative rings and in algebraic geometry, but may appear in any investigation of the structure of a ring as a result of the Yoneda embedding which sends a ring to the category of left modules over that ring.