Noether normalization
Every finitely generated algebra over a field, no matter how tangled its defining relations, can be presented as a modest amount of 'thickening' sitting on top of clean coordinate axes. Noether normalization makes this precise: it finds inside your algebra a polynomial ring — a piece of pure, relation-free coordinate space — over which the whole algebra is only a finite extension. Geometrically, every affine variety projects, finitely and surjectively, onto an affine space of the same dimension.
The lemma states: if A is a finitely generated algebra over a field k, then there exist elements y_1, ..., y_d in A that are algebraically independent over k such that A is a finitely generated module over the polynomial subring k[y_1, ..., y_d]. Equivalently A is integral over this polynomial ring. The number d is forced — it equals the Krull dimension of A and the transcendence degree of its fraction field over k.
The proof, in characteristic zero or over an infinite field, uses a generic linear change of coordinates to make one generator integral over the rest, then induction; over finite fields one uses nonlinear substitutions instead. This lemma is the engine behind a clean proof of the Nullstellensatz, behind the dimension theory of varieties, and behind the finiteness statements of algebraic geometry.
For A = k[x, y]/(xy - 1), the element t = x is transcendental and A = k[x, x^{-1}] is finite over... actually take t = x + y: then A is integral over k[t], realizing the hyperbola xy = 1 as a finite cover of the t-line of dimension 1.
A curve presented as finite over a one-dimensional coordinate line.
Combined with going-up, normalization shows dim A = d directly, since A is integral over k[y_1, ..., y_d] which has dimension d. It is the algebraic backbone of the statement 'dimension = number of free parameters'.