projective variety
Affine varieties have a frustrating habit of 'running off to infinity': two parallel lines never meet, a hyperbola has open ends. Projective geometry adds an extra rim of points at infinity so that these escaping pieces are caught. A projective variety lives in that compactified world, where lines always meet and curves close up.
Concretely, projective n-space P^n has homogeneous coordinates (x_0 : ... : x_n), well defined only up to a common nonzero scalar. A polynomial cannot be evaluated at such a point, but its vanishing is meaningful when the polynomial is homogeneous, since scaling all coordinates by t multiplies the value by t^d. A projective variety is the common zero locus in P^n of a collection of homogeneous polynomials.
The payoff is dramatic: projective varieties over an algebraically closed field are complete (the analogue of compactness), so images of morphisms are closed and intersection numbers behave well, as in Bezout's theorem. The trade is that there is no single global coordinate ring of regular functions; the only globally regular functions on an irreducible projective variety are the constants, so one must work with homogeneous ideals and graded rings instead.
In P^2 the homogeneous cubic V(y^2 z - x^3 - a x z^2 - b z^3) is a plane elliptic curve; its single point at infinity (0 : 1 : 0) serves as the identity of the group law.
Adding the point at infinity is exactly what makes the elliptic-curve group law work.
A homogeneous ideal that vanishes only at the origin of A^{n+1} (the so-called irrelevant ideal, generated by all the variables) defines the empty projective variety, since the origin is excluded from P^n.