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Homology with coefficients in an abelian group G, denoted H_n(X; G), is obtained by tensoring the singular chain complex with G: C_n(X; G) = C_n(X) tensor G, with boundary operators d tensor id_G. The choice of coefficient group G dramatically affects the resulting homology: integer coefficients Z give the most information but are hardest to compute; field coefficients (Q, Z/pZ) eliminate torsion subtleties and make homology a vector space; Z/2Z coefficients are natural for non-orientable manifolds and mod-2 intersection theory. The universal coefficient theorem relates H_n(X; G) back to H_n(X; Z).
Homology with coefficients generalizes singular homology by replacing the integers Z with an arbitrary abelian group G. The construction is straightforward: define C_n(X; G) = C_n(X; Z) tensor_Z G. Since C_n(X; Z) is a free abelian group (generated by singular n-simplices), the tensor product C_n(X; Z) tensor G is the free G-module on the same generators. A chain is now a formal finite sum of singular simplices with coefficients in G (instead of Z). The boundary operator d tensor id : C_n(X; G) -> C_{n-1}(X; G) is the same alternating sum formula, applied coefficientwise. The resulting homology H_n(X; G) = ker(d_n)/im(d_{n+1}) in this new chain complex.
The most important coefficient choices are: Z (integer coefficients, the default), Q (rational coefficients, which kill all torsion), Z/pZ for a prime p (mod-p coefficients, which detect p-primary torsion), and R or C (real or complex coefficients, used in connection with de Rham theory and Hodge theory). Each choice provides a different "lens" on the topology: rational homology sees only the free part of the integer homology (the Betti numbers), while mod-p homology detects p-primary torsion with high precision. The philosophy is that no single coefficient group captures everything — one should compute homology with various coefficients and combine the results.
The universal coefficient theorem for homology relates H_n(X; G) to H_*(X; Z): there is a short exact sequence 0 -> H_n(X; Z) tensor G -> H_n(X; G) -> Tor(H_{n-1}(X; Z), G) -> 0, which splits (non-naturally). The tensor term is the "expected" contribution: each Z summand in H_n(X; Z) contributes a copy of G, and each Z/mZ summand contributes G/mG. The Tor term is the correction from torsion in one degree lower: each Z/mZ summand in H_{n-1}(X; Z) contributes Tor(Z/mZ, G) to H_n(X; G). For G = Z/pZ: Tor(Z/mZ, Z/pZ) = Z/gcd(m,p)Z, which is Z/pZ when p divides m and 0 otherwise.
Field coefficients are technically simpler because Tor vanishes for fields (every module over a field is flat). With field coefficients k, homology groups are vector spaces: H_n(X; k) = H_n(X; Z) tensor k, with dimension equal to the free rank of H_n(X; Z) (over Q) or the free rank plus a correction for the relevant torsion (over Z/pZ). The Euler characteristic and Betti numbers are most naturally defined using field coefficients, and many theorems (Poincare duality with Z/2Z, the Kunneth formula over fields) take their simplest form with field coefficients.
In practice, the most common workflow is: compute H_*(X; Z) directly if possible (using cellular or simplicial methods), then derive H_*(X; G) for other G via the universal coefficient theorem. Alternatively, for spaces where the integer computation is hard, one may compute H_*(X; Q) (to get Betti numbers) and H_*(X; Z/pZ) for small primes p (to detect torsion), then reconstruct H_*(X; Z) from these partial results. The universal coefficient theorem ensures no information is lost in this process — the integer homology is uniquely determined by the collection of all mod-p homologies together with the rational homology.