height of a prime
If Krull dimension counts how tall a tower of primes can be in the whole ring, the height of a single prime counts how far down you can dig below that prime. Geometrically, a prime ideal corresponds to an irreducible subvariety, and its height is the codimension — how many independent conditions you imposed to cut that subvariety out of the ambient space. A point in 3-space has height 3; a surface in it, height 1.
The height of a prime ideal p in a commutative ring R is the supremum of lengths of chains of primes p_0 ⊊ p_1 ⊊ ... ⊊ p_n = p descending from p. Equivalently it is the Krull dimension of the localization R_p. Minimal primes have height 0; in an integral domain the zero ideal is the unique minimal prime, of height 0.
Height and dimension fit together but need not add up cleanly. For a prime p one always has height(p) + dim(R/p) <= dim(R), with equality for finitely generated algebras over a field that are domains (these are 'catenary' and well-behaved). In pathological rings equality can fail and even height itself can be infinite, so the clean codimension intuition is licensed only in the geometric, finitely generated setting.
In k[x, y, z], the prime (x, y, z) has height 3 (chain (0) ⊊ (x) ⊊ (x, y) ⊊ (x, y, z)), the prime (x, y) has height 2, and (x) has height 1 — heights matching codimensions of point, line, plane in 3-space.
Heights in k[x, y, z] equal codimensions.
Krull's principal ideal theorem bounds height by generators: a prime minimal over an ideal generated by r elements has height at most r. This is the algebraic form of 'each equation cuts dimension by at most one'.