An affine variety is an algebraic variety contained in affine space. For example,
(1)

is the cone, and
(2)

is a conic section, which is a subvariety of the cone. The cone can be written to indicate that it is the variety corresponding to . Naturally, many other polynomials vanish on , in fact all polynomials in . The set is an ideal in the polynomial ring . Note also, that the ideal of polynomials vanishing on the conic section is the ideal generated by and .
A morphism between two affine varieties is given by polynomial coordinate functions. For example, the map is a morphism from to . Two affine varieties are isomorphic if there is a morphism which has an inverse morphism. For example, the affine variety is isomorphic to the cone via the coordinate change .
Many polynomials may be factored, for instance , and then . Consequently, only irreducible polynomials, and more generally only prime ideals are used in the definition of a variety. An affine variety is the set of common zeros of a collection of polynomials , ..., , i.e.,
(3)

as long as the ideal is a prime ideal. More classically, an affine variety is defined by any set of polynomials, i.e., what is now called an algebraic set. Most points in will have dimension , but may have singular points like the origin in the cone.
When is onedimensional generically (at almost all points), which typically occurs when , then is called a curve. When is twodimensional, it is called a surface. In the case of CWcomplex affine space, a curve is a Riemann surface, possibly with some singularities.
The Wolfram Language function ContourPlot will graph affine varieties in the real affine plane. For example, the following graphs a hyperbola and a circle.
GraphicsGrid[{{ ContourPlot[x^2  y^2 == 1, {x, 2, 2}, {y, 2, 2}], ContourPlot[x^2 + y^2 == 1, {x, 2, 2}, {y, 2, 2}] }}]