the structure sheaf
Once you have decided that the points of a ring R should be its prime ideals (the space Spec R), you still need to put functions on that space — otherwise it is just a bare topological set with no geometry. The structure sheaf O_{Spec R} is the sheaf of rings that supplies exactly these functions, turning the point-set Spec R into a genuine geometric object. It is the precise device that makes 'an element of R' into a global function and, more importantly, tells you what the functions on each open subset are.
The construction starts on the distinguished opens. On D(f) = {primes not containing f}, the structure sheaf is the localization R_f = R[1/f] — you are allowed to divide by f and its powers, exactly because f is invertible (nonzero) on D(f). In particular the global sections O_{Spec R}(Spec R) = R recover the whole ring, and the sections over the basic open D(f) are R_f. On a general open set these assignments are glued together by the sheaf axioms — formally, the value O_{Spec R}(U) is the inverse limit of R_f over the basic opens D(f) inside U. The crucial payoff is the stalk: at a prime p, the stalk O_{Spec R, p} is the localization R_p (invert everything outside p), which is a LOCAL ring with maximal ideal pR_p. So (Spec R, O_{Spec R}) is a locally ringed space, and that is the definition of an affine scheme.
The structure sheaf is what carries all the geometry: regular functions, the local rings recording infinitesimal behavior at each point, and later the entire theory of sheaves of O_X-modules (quasi-coherent and coherent sheaves) live over it. One subtle point worth stating honestly: the value of the structure sheaf on a basic open D(f) is the localization R_f, NOT simply 'the ring R restricted to D(f)' in any naive set-theoretic sense — sections over an open are functions that may have poles outside it, captured by inverting f. And because R may have nilpotents, two different rings can have homeomorphic spectra yet non-isomorphic structure sheaves (e.g. k[x] and k[x]/(x^2) both have one or a couple of points but very different O_X), which is exactly why the structure sheaf, not the underlying space, is the real content of a scheme.
For R = k[x], the affine line, the global sections of O_X are the whole polynomial ring k[x]. On the basic open D(x) = {x not equal to 0} the sections are k[x, 1/x] (Laurent polynomials — you may now divide by x), and the stalk at the point (x - a) is the local ring k[x]_{(x-a)} of rational functions with no pole at a. Inverting f literally means 'allow poles only outside D(f)'.
On the affine line, O_X(D(f)) inverts f: sections may have poles exactly where f vanishes.
Sections over D(f) are the localization R_f, not a naive restriction — they may have poles off D(f). And the structure sheaf, not the topological space, is the real data: rings with nilpotents (k[x]/(x^2)) have a tiny space but a structure sheaf remembering the nilpotent.