chain complex
Imagine a row of rooms connected by one-way doors, where each door leads to the next room and then to a room marked “empty”. The rule of the building is that if you walk through two doors in a row, you always end up in the empty room. A chain complex packages exactly this rule for modules: a sequence of arrows where applying two arrows in a row gives zero. That single condition is what makes the whole theory of homology possible.
Formally, a chain complex over a ring R is a sequence of R-modules and homomorphisms ... -> C_{n+1} -> C_n -> C_{n-1} -> ... with maps d_n : C_n -> C_{n-1} (the boundary maps, or differentials) satisfying d_n ∘ d_{n+1} = 0 for every n. The condition d^2 = 0 is equivalent to saying that the image of d_{n+1} is contained in the kernel of d_n. Elements of ker(d_n) are called n-cycles and elements of im(d_{n+1}) are called n-boundaries; every boundary is a cycle.
The complex is exact at C_n precisely when im(d_{n+1}) equals ker(d_n) — that is, when there are no cycles beyond the obvious ones. Homology measures how badly this fails. The indexing here is homological (degrees decrease along the arrows); the same data with degrees increasing is called a cochain complex. A chain complex of vector spaces, abelian groups, or modules over any ring is the basic object on which derived functors, resolutions, and spectral sequences are built.
Over Z take ... -> 0 -> Z -> Z -> 0 with the single nonzero map being multiplication by 2: d(x) = 2x. Then d^2 = 0 holds vacuously since the next map is zero. This two-term complex resolves Z/2Z.
A short complex whose only differential is multiplication by 2.
A chain map between two complexes is a family of maps commuting with the differentials; such maps preserve cycles and boundaries, so they descend to maps on homology. The category of chain complexes over R is itself abelian, which is why homological constructions can be iterated.