Krull dimension
How many independent directions does a geometric object have — is it a point, a curve, a surface, a solid? Krull dimension answers this purely algebraically, without any picture, by counting how deeply prime ideals can be nested. A point's ring has no room to nest; a curve's ring lets you nest one prime inside another; a surface's lets you nest two. The longest tower of primes you can build measures the dimension of the space.
The Krull dimension of a commutative ring R is the supremum of lengths n of chains of distinct prime ideals p_0 ⊊ p_1 ⊊ ... ⊊ p_n. (A chain with n+1 primes has length n.) A field has dimension 0; the integers Z and any PID that is not a field have dimension 1 (chains (0) ⊊ (p)); and the polynomial ring k[x_1, ..., x_n] over a field has dimension n.
For finitely generated algebras over a field this combinatorial count agrees with every geometric notion of dimension, thanks to Noether normalization, and with the transcendence degree of the function field. But in full generality Krull dimension can misbehave: there are Noetherian rings whose dimension is infinite (Nagata's examples), so finiteness of dimension is a real theorem (true for finitely generated algebras over a field or over Z), not an automatic fact.
In k[x, y] the chain (0) ⊊ (x) ⊊ (x, y) of primes has length 2 and is maximal, so dim k[x, y] = 2 — matching the plane being two-dimensional.
A maximal prime chain in the polynomial ring of the plane.
Krull dimension of a local ring equals the height of its maximal ideal, and for Noetherian local rings it also equals the minimal number of generators of an ideal whose radical is the maximal ideal — the dimension theorem of Krull.