dimension of a variety
A point is zero-dimensional, a curve one-dimensional, a surface two-dimensional — our intuition about dimension is clear for the smooth examples. Algebraic geometry must define dimension purely algebraically, so that it makes sense over any field, at singular points, and even for schemes with no underlying real picture.
The chosen definition is the Krull dimension of the coordinate ring: the supremum of lengths n of strictly increasing chains of prime ideals p_0 ⊂ p_1 ⊂ ... ⊂ p_n in k[X]. Geometrically this counts the longest chain of nested irreducible subvarieties, since primes correspond to irreducible closed sets. For an irreducible variety it equals the transcendence degree over k of the function field, the field of fractions of k[X].
Several other measures agree for nice (finite-type, irreducible) varieties: the dimension equals the length of any maximal chain of irreducible subvarieties (the ring is catenary), and at a smooth point it equals the vector-space dimension of the tangent space. At singular points the tangent space can be strictly larger, which is one clean test for smoothness. Dimension is local on irreducible components but a reducible variety takes the maximum over its components.
Affine n-space A^n has dimension n: the chain (0) ⊂ (x_1) ⊂ (x_1, x_2) ⊂ ... ⊂ (x_1, ..., x_n) of primes in k[x_1, ..., x_n] has length n, and the function field k(x_1, ..., x_n) has transcendence degree n.
Two computations, one answer n — Krull length and transcendence degree coincide.