singular homology
How many holes does a space have, and of what dimension? A circle has one one-dimensional hole; a sphere has a two-dimensional cavity but no one-dimensional hole; a torus has two one-dimensional holes and one two-dimensional one. Singular homology is the machine that turns these vague counts into honest abelian groups H_n(X), one for each dimension n, defined for absolutely any topological space without needing the space to be triangulated or built nicely.
Here is the construction in plain steps. A singular n-simplex is just a continuous map sigma: Delta^n -> X from the standard n-simplex (a point, a segment, a triangle, a tetrahedron, and so on) into X. Free abelian groups C_n(X) are built from formal integer combinations of these maps — these are the n-chains. A boundary operator d_n: C_n -> C_{n-1} sends each simplex to the alternating sum of its faces, and the crucial identity d_{n-1} d_n = 0 holds. So boundaries (the image of d_{n+1}) sit inside cycles (the kernel of d_n), and the n-th homology group is the quotient H_n(X) = ker(d_n) / im(d_{n+1}): cycles that are not boundaries, counted up to the relation 'differs by a boundary'.
The payoff is that H_n is a homotopy invariant (homotopy-equivalent spaces have isomorphic homology) and is computable in practice through long exact sequences, excision, and cellular methods, even though the chain groups C_n(X) themselves are enormous (uncountably generated). You almost never compute with the singular chains directly; their value is theoretical — they make the theory defined for every space and manifestly functorial, so a continuous map f: X -> Y induces homomorphisms f_*: H_n(X) -> H_n(Y) automatically.
An honest caveat: homology counts holes only up to abelianization. It cannot see non-commutative information that the fundamental group records — H_1(X) is exactly the abelianization of pi_1(X), so it forgets the order in which loops are traversed. Two spaces can have identical homology in every degree yet be genuinely different (and even non-homotopy-equivalent), so homology is a powerful but lossy invariant.
For the 2-sphere S^2: H_0 = Z (one connected component), H_1 = 0 (no one-dimensional holes — every loop on S^2 bounds a cap), and H_2 = Z (one two-dimensional cavity, generated by the fundamental cycle of the whole sphere). All higher H_n vanish. Contrast the torus T^2: H_0 = Z, H_1 = Z + Z (two independent loops), H_2 = Z.
Homology of S^2 versus T^2: H_1 distinguishes them at a glance.
H_1(X) is the abelianization of pi_1(X), so homology is strictly weaker than the fundamental group in degree 1: it cannot tell apart spaces whose loops commute the same way but interact non-abelianly.