completion
Completion is the algebraic version of zooming infinitely far in and allowing infinite Taylor expansions. Localizing already focuses you near a point; completion goes the final step, replacing polynomial behavior by full power-series behavior, so that every Cauchy-like sequence of approximations actually converges to something in the ring. It is the move from polynomials k[x] to formal power series k〚x〛, or from the integers Z to the p-adic integers Z_p.
Given a ring R and an ideal I, the I-adic completion is the inverse limit of the system of quotients R -> ... -> R/I^3 -> R/I^2 -> R/I. An element is a compatible sequence of residues modulo each power of I — a 'power series in I'. The natural map R -> R-hat is injective exactly when the intersection of all I^n is zero (Krull intersection); for a Noetherian local ring with I = m the maximal ideal, completion gives the m-adic completion R-hat, again local with maximal ideal generated by the image of m.
Completion is a flat, exact, and structure-preserving operation that keeps dimension and regularity (R regular iff R-hat regular). Its power is that completed local rings are far more rigid: the Cohen structure theorem says a complete regular local ring containing a field is just a power series ring k〚x_1, ..., x_d〛. The trade-off is that completion forgets global, far-away information entirely — it sees only an infinitesimal neighborhood, so two very different varieties can have isomorphic completions at corresponding smooth points.
The (p)-adic completion of Z is Z_p = inverse limit of Z/p^n Z; an element is a coherent sequence (a_1 mod p, a_2 mod p^2, ...). Likewise the (x)-adic completion of k[x] is the power series ring k〚x〛.
Z completes to the p-adic integers; k[x] to power series.
Hensel's lemma holds in complete local rings: an approximate root of a polynomial can be refined to an exact one, much like Newton's method. This is why completions are central in number theory and in lifting solutions modulo p^n.