localization at a prime
Geometry is local: to understand a curve near a point, you zoom in and ignore everything happening elsewhere. Localizing at a prime is the algebraic zoom lens. You take a prime ideal p — think of it as a point — and you make invertible every function that does not vanish there, so that all that survives is the behavior of the ring infinitesimally close to that single point. The result is a local ring, with exactly one maximal ideal: the point you focused on.
Given a prime ideal p of a commutative ring R, the complement S = R \ p is a multiplicative set (a, b not in p implies ab not in p, precisely because p is prime). The localization R_p is the ring of fractions S^{-1}R, consisting of formal fractions a/s with a in R and s not in p. The ideal pR_p is the unique maximal ideal of R_p, so R_p is a local ring, and its residue field R_p / pR_p is the field of fractions of R/p.
Localization at a prime is exact and well-behaved: prime ideals of R_p correspond bijectively to primes of R contained in p, so it 'remembers only what lies below p'. The honest caveat is that this is not finite — R_p is usually much bigger than R as a set of formal fractions — but it is the right tool for stalk-by-stalk arguments, since a property like being zero, or a sequence being exact, can be checked locally at every prime.
Localizing Z at the prime (p) gives Z_(p) = { a/b : p does not divide b }, the rational numbers with denominators prime to p. Its unique maximal ideal is pZ_(p), and the residue field is Z/pZ.
Z localized at (p): the local ring of the integers at one prime.
The dimension of R_p equals the height of p, and the local rings R_p for all primes p are the stalks of the structure sheaf on Spec(R) — localization is exactly how schemes are glued from local pieces.