affine variety
Think of a curve like a circle x^2 + y^2 = 1 sitting in the plane. It is the solution set of a polynomial equation. An affine variety is exactly this idea pushed to its limit: take any collection of polynomials in several variables, and look at all the points where they simultaneously vanish. Geometry (the shape) and algebra (the equations) become two views of one object.
Precisely, fix a field k and consider affine n-space A^n over k. For a set S of polynomials in k[x_1, ..., x_n], the affine variety V(S) is the set of points a in A^n with f(a) = 0 for every f in S. By Hilbert's basis theorem every such S is contained in a finitely generated ideal, so finitely many polynomials already cut out V(S). Many authors reserve the word 'variety' for the irreducible case and call the general object an affine algebraic set.
There is a caveat about the ground field. Over an algebraically closed field the Nullstellensatz makes the dictionary between varieties and radical ideals perfectly tight, so the points really 'see' all the algebra. Over a non-closed field like the reals, the equation x^2 + 1 = 0 has empty point set yet a nontrivial ideal, and the scheme-theoretic viewpoint is later introduced precisely to repair such mismatches.
Over C, the variety V(y - x^2) in A^2 is the parabola; V(xy) is the union of the two coordinate axes, hence reducible; V(1) is empty and V(0) is all of A^2.
One smooth curve, one reducible pair of lines, and the two trivial extremes.
Terminology varies: in some texts 'affine variety' already means an irreducible affine algebraic set, while others allow reducible ones. Always check whether irreducibility is built into the author's definition.