a Chern class
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A Chern class is a number-valued (more precisely cohomology-valued) measure of how twisted a complex vector bundle is. The first Chern class of a complex line bundle counts, in a topological sense, how the fiber rotates as you wander around the base — over a sphere it is an integer, the same integer that counts the windings of the gluing or the degree of the bundle. Higher Chern classes do the analogous job for higher-rank bundles. They are the most important invariants for distinguishing complex bundles.
Precisely, to a complex vector bundle E of rank k over M one associates Chern classes c_1(E), c_2(E), ..., c_k(E) with c_j(E) in H^{2j}(M; Z), packaged into the total Chern class c(E) = 1 + c_1 + ... + c_k. They are characterized by axioms: naturality under pullback, the Whitney sum formula c(E + F) = c(E) c(F), normalization fixing c_1 of the tautological line bundle on CP^1, and c_j = 0 above the rank. Via Chern-Weil they have explicit curvature representatives: choosing any connection with curvature Omega, the total Chern class is represented by det(I + (i/2pi) Omega), so c_1 is represented by (i/2pi) tr(Omega) and c_2 by the next symmetric function — concrete closed forms whose integrals are the integer invariants. The first Chern class c_1 is the single most-used: it classifies complex line bundles completely (c_1: line bundles -> H^2(M; Z) is an isomorphism).
Chern classes are everywhere in geometry and physics: c_1 of the canonical bundle defines a complex manifold's first Chern class (Calabi-Yau means c_1 = 0), the index theorem expresses analytic indices as integrals of Chern classes (the Chern character), and in physics c_1 of a line bundle is the quantized magnetic flux of a monopole. One honest caveat: the Chern-Weil curvature formulas compute the image of the Chern classes in real cohomology, so they capture the classes up to torsion; the genuinely integral Chern class, which can have torsion, requires the topological definition. Also, c_j lives in degree 2j, not degree j — a frequent indexing slip.
The tautological line bundle O(-1) over CP^1 (the 2-sphere) has c_1 equal to the negative generator of H^2(CP^1; Z) = Z, so integral over CP^1 of c_1 = -1. Its dual, the hyperplane bundle O(1), has c_1 integrating to +1. By Chern-Weil you can compute this with the Fubini-Study connection: (i/2pi) tr(Omega) is the Fubini-Study area form scaled to total area 1, and integrating gives exactly the integer +1 — independent of which connection you used.
Over CP^1 the line bundles are labeled by a single integer, their c_1 — the degree.
The j-th Chern class lives in degree 2j (so c_1 in H^2, c_2 in H^4), and the Chern-Weil curvature formulas give only the real image, missing torsion. Also, the total Chern class is multiplicative under Whitney sum (c(E + F) = c(E)c(F)), but the individual c_j are not additive — c_1(E+F) = c_1(E) + c_1(F) holds, but c_2 picks up a cross term c_1(E)c_1(F).