a holomorphic line bundle
Attach a one-dimensional complex line (a copy of C) to every point of a complex manifold, smoothly twisting from point to point, with the twisting recorded by holomorphic gluing rules. That is a holomorphic line bundle. The cylinder (no twist) and the Mobius band (one twist) are the cartoon real pictures; over a complex manifold the line is complex and the gluing functions are nonvanishing holomorphic functions. Line bundles are the indispensable vocabulary for talking about 'functions that are not quite functions' — objects like meromorphic functions, theta functions, and sections of a positive bundle that you embed a variety with.
Precisely, a holomorphic line bundle L over a complex manifold M is given by an open cover U_a and holomorphic transition functions g_{ab}: U_a intersect U_b -> C^* (nonzero complex numbers) satisfying the cocycle condition g_{ab} g_{bc} = g_{ac} on triple overlaps. A holomorphic section is a holomorphic choice of a point in each fibre — locally a holomorphic function s_a with s_a = g_{ab} s_b on overlaps. The set of holomorphic line bundles forms a group under tensor product (the dual bundle is the inverse, the trivial bundle the identity); this group is the Picard group Pic(M). On a compact manifold the most important invariant of L is its first Chern class c_1(L) in H^2(M, Z), which measures the bundle's topological twist and, together with a metric, its curvature.
Line bundles are where the action is because global holomorphic functions on a compact manifold are only constants — so to get a rich supply of holomorphic objects you allow controlled poles and zeros, which is exactly what sections of a line bundle encode. The dimension of the space of sections, h^0(M, L), and its higher cohomology are computed by Riemann-Roch and controlled by vanishing theorems. A frequent confusion: a line bundle's sections are not functions on M (they live in the twisted bundle), and a nontrivial line bundle may have no nonzero global holomorphic sections at all (e.g. a bundle of negative degree on a curve).
Over CP^n the tautological line bundle O(-1) assigns to each point (a line through the origin in C^{n+1}) that very line as its fibre; its dual O(1) is the hyperplane bundle. The global holomorphic sections of O(k) for k >= 0 are exactly the degree-k homogeneous polynomials in the n+1 coordinates, and O(1) is the positive bundle whose sections give the embedding of CP^n into a bigger projective space.
On CP^n, sections of O(k) are degree-k homogeneous polynomials; O(1) is the positive (ample) bundle.
Sections of a line bundle are not functions on M, and a bundle can have zero global sections. The number of sections is a subtle cohomological quantity governed by Riemann-Roch and vanishing theorems, not something you can read off the topology alone.