module
A module is what you get when you take the idea of a vector space and let the scalars come from a ring instead of a field. You still have vectors you can add and scale, but now the scalars need not have inverses — you can multiply a vector by 2, but you may not be able to divide by 2. Most of the comfortable behaviour of vector spaces (bases, dimension, complements) becomes optional, and which of it survives is exactly what module theory studies.
Precisely, let R be a ring with 1. A left R-module is an abelian group (M, +) together with a scalar multiplication R × M -> M, written (r, m) |-> rm, satisfying r(m + n) = rm + rn, (r + s)m = rm + sm, (rs)m = r(sm), and 1m = m for all r, s in R and m, n in M. Right modules are defined symmetrically with scalars acting on the right. When R is commutative the two notions coincide.
Modules unify a remarkable range of objects. A module over a field is exactly a vector space; a module over Z is exactly an abelian group (the scalar action is forced: n times m is m + m + ... + m); a module over the polynomial ring k[x] is a vector space together with a chosen linear operator (x acts as that operator). This last viewpoint turns linear algebra over k[x] into the theory of a single operator, and the structure theorem then recovers Jordan and rational canonical forms.
Z/6Z is a Z-module: you can add elements and multiply by integers, but 2 has no inverse and the element 3 satisfies 2·3 = 0 even though neither factor is zero — behaviour impossible in a vector space.
An abelian group is just a Z-module; torsion appears the moment scalars stop being invertible.
The requirement 1m = m is called unitality and is almost always assumed; without it one can have a stray direct summand on which R acts as zero. Over a field, the failure of scalars to be invertible disappears, and that single fact is why every vector space has a basis but most modules do not.