an affine variety

Think of the points in the plane where a polynomial vanishes: x^2 + y^2 - 1 = 0 carves out a circle, y - x^2 = 0 carves out a parabola. An affine variety is exactly this picture taken seriously and in any number of variables: the set of all points in affine space A^n where a chosen collection of polynomials simultaneously equal zero. It is the geometric shadow cast by a system of polynomial equations.

Precisely, fix an algebraically closed field k (you may keep the complex numbers C in mind) and let A^n be n-dimensional affine space, just the set of n-tuples (a_1, ..., a_n) with a_i in k, with no chosen origin or linear structure singled out. Given any set S of polynomials in k[x_1, ..., x_n], its zero locus V(S) = { p in A^n : f(p) = 0 for all f in S } is called an algebraic set. Many authors reserve the word variety for an algebraic set that is irreducible, meaning it cannot be written as a union of two strictly smaller algebraic sets; a circle is irreducible, but the pair of lines xy = 0 is not. By Hilbert's basis theorem any S is captured by finitely many polynomials, so every variety is cut out by a finite system.

Affine varieties are the ground floor of algebraic geometry: projective varieties are glued from them, schemes generalize them, and the entire ideal-variety dictionary (the Nullstellensatz, the coordinate ring, dimension) is built on this object. One honest caveat: the field must be algebraically closed for the geometry to match the algebra. Over R the equation x^2 + y^2 + 1 = 0 has no real points at all, yet it is a perfectly good nonzero ideal; only over an algebraically closed field do nonzero proper ideals always carry points, which is precisely what the Nullstellensatz guarantees.