an invertible sheaf and the Picard group
Among all sheaves of modules on a scheme, the simplest non-trivial ones are the 'line bundles' — sheaves that locally look like a single copy of the structure sheaf, O_X over each small open set, but which may twist nontrivially as you move around the whole space. An invertible sheaf is the scheme-theoretic name for a line bundle, and the collection of all of them, with a natural multiplication, forms the Picard group — a fundamental invariant measuring how many essentially different ways the space can be 'twisted' by a one-dimensional bundle.
Precisely, an invertible sheaf L on a scheme X is a locally free O_X-module of rank one: every point has an open neighborhood U on which L restricted to U is isomorphic to O_X restricted to U. The name 'invertible' is exact: L is invertible if and only if there is a sheaf L^{-1} (its dual, Hom(L, O_X)) with L tensor L^{-1} isomorphic to O_X — under tensor product, invertible sheaves have inverses. So the set of isomorphism classes of invertible sheaves, with tensor product as the group operation, O_X as identity, and the dual as inverse, forms an abelian group: the Picard group Pic(X). Computing Pic(X) is computing how many line bundles X carries up to isomorphism — for projective space Pic(P^n) = Z, generated by the tautological twisting sheaf O(1), so every line bundle on P^n is O(d) for a unique integer d, the degree.
The Picard group is one of the central invariants of a scheme and the bridge to divisor theory: on a sufficiently nice (integral, Noetherian, separated, normal) scheme there is an isomorphism between Pic(X) and the group of Cartier divisors modulo principal divisors, so 'line bundle up to iso' and 'divisor up to linear equivalence' are two names for the same thing. This is the line-bundle/divisor correspondence that powers Riemann-Roch. Honest cautions. First, invertible sheaves are exactly the rank-one LOCALLY FREE sheaves — a rank-one coherent sheaf that is not locally free (an ideal sheaf of a point, say) is NOT invertible, so 'rank one' alone is insufficient. Second, the correspondence with divisors needs hypotheses: on a singular or non-normal scheme, Cartier divisors (locally principal, hence always tied to line bundles) and Weil divisors (codimension-one cycles) can DIFFER, and Pic matches only the Cartier side — over a smooth variety they agree, but never assume Weil = Cartier in general.
On P^1 the line bundle O(1) is the dual of the tautological subbundle; its global sections are the linear forms a x + b y, a two-dimensional space. Tensoring, O(d) = O(1)^{tensor d} has global sections the degree-d homogeneous polynomials (dimension d + 1 for d >= 0, and 0 for d < 0). Every invertible sheaf on P^1 is some O(d), so Pic(P^1) = Z.
On P^1, line bundles are exactly the O(d), so Pic(P^1) = Z with O(1) the generator.
Invertible = rank-one LOCALLY FREE, not merely rank one: an ideal sheaf has generic rank one but is not invertible. And Pic matches Cartier divisors modulo principal — on singular schemes Weil and Cartier divisors differ, so do not equate Pic with the Weil divisor class group.