ideal of a variety
Given a shape sitting in space, you can ask: which polynomial equations does it satisfy? Collect every polynomial that vanishes identically on the shape, and you get the ideal of the variety. It is the algebraic shadow of the geometry, the reverse direction of the map that sends an ideal to its vanishing set.
Formally, for a subset X of affine n-space, I(X) is the set of f in k[x_1, ..., x_n] with f(a) = 0 for all a in X. It is an ideal — sums and multiples of vanishing polynomials still vanish — and it is always a radical ideal, since if f^m vanishes on X then so does f. The operators V (vanishing set) and I (ideal of) form an order-reversing Galois connection between ideals and subsets.
How tightly do V and I undo each other? Always I(V(J)) contains the radical of J, and V(I(X)) is the Zariski closure of X. Hilbert's Nullstellensatz, over an algebraically closed field, makes the first inclusion an equality: I(V(J)) equals the radical of J. Thus radical ideals correspond exactly to Zariski-closed sets, the precise statement of the algebra-geometry dictionary that fails over non-closed fields.
I(V(x^2)) = (x), not (x^2): the doubled point x^2 = 0 has the same point set as x = 0, so passing to vanishing-then-ideal loses the multiplicity and returns the radical.
The classical I-V dictionary cannot see multiplicities — one motivation for scheme theory.