the coordinate ring
Geometry and algebra are two views of the same thing, and the coordinate ring is the device that turns a variety into an algebra you can compute with. Given a variety, its coordinate ring is the ring of all polynomial functions on it — what you get when you start with all polynomials and then declare two of them equal whenever they agree at every point of the variety. The shape of the variety is fully encoded in the algebraic structure of this ring.
For an affine variety X in A^n with ideal I(X), the coordinate ring is the quotient k[X] = k[x_1, ..., x_n] / I(X). An element is a polynomial restricted to X, with two polynomials identified exactly when they differ by something in I(X), i.e. by a function vanishing on all of X. Because I(X) is a radical ideal, k[X] is a reduced ring (no nonzero nilpotents); it is also a finitely generated k-algebra. The dictionary is tight and contravariant: points of X correspond to maximal ideals of k[X]; irreducible closed subvarieties correspond to prime ideals; X is irreducible exactly when I(X) is prime, equivalently when k[X] is an integral domain. Morphisms of varieties X -> Y correspond, reversing direction, to k-algebra homomorphisms k[Y] -> k[X].
This anti-equivalence — affine varieties on one side, reduced finitely generated k-algebras on the other — is the engine of the whole subject. It lets you replace a geometric question by a ring-theoretic one and back. Dimension becomes Krull dimension of the ring, smoothness becomes a regularity condition, and the leap to schemes comes from simply allowing arbitrary commutative rings, including non-reduced ones, on the algebra side. Caveat: this clean story is for affine varieties. A projective variety's homogeneous coordinate ring is graded and depends on the chosen embedding, and its only honest global functions are constants, so the coordinate ring is not the right invariant in the projective setting — there you work locally and with the function field.
For the parabola X = V(y - x^2) in A^2, the coordinate ring is k[x, y] / (y - x^2). Since y is forced to equal x^2, every function reduces to a polynomial in x alone, so k[X] is isomorphic to k[x]: the parabola is, as an abstract variety, just the affine line in disguise.
The coordinate ring sees the intrinsic variety, not the embedding: a parabola and a line have isomorphic coordinate rings.
The clean affine-variety/algebra dictionary needs a reduced, finitely generated algebra over an algebraically closed field. For projective varieties the homogeneous coordinate ring depends on the embedding and is not the intrinsic invariant.