Nullstellensatz
The Nullstellensatz — German for 'theorem of zeros' — is the bridge that makes algebra and geometry two views of one thing. On the algebra side you have ideals of polynomials; on the geometry side you have the sets where those polynomials vanish. The theorem says these two worlds are perfectly mirrored: questions about shapes become questions about ideals, and vice versa, with no loss of information, provided you work over an algebraically closed field.
Let k be an algebraically closed field. The weak Nullstellensatz says every maximal ideal of k[x_1, ..., x_n] has the form (x_1 - a_1, ..., x_n - a_n) for a point (a_1, ..., a_n), so a system of polynomials with no common zero must generate the whole ring (1 lies in the ideal). The strong Nullstellensatz says that if a polynomial f vanishes on the common zero set V(I) of an ideal I, then some power f^m lies in I; equivalently I(V(I)) = sqrt(I). Thus radical ideals correspond bijectively, and inclusion-reversingly, to affine algebraic sets.
The hypothesis that k is algebraically closed is essential: over the reals, x^2 + 1 has no zeros yet generates a proper ideal of R[x], breaking the weak form. The strong form's appearance of the radical is exactly why radical ideals, not arbitrary ideals, are the correct geometric objects — many ideals can define the same vanishing set, and only their common radical is seen by geometry.
Over C, the ideals (x) and (x^2) in C[x, y] have the same vanishing set, the line x = 0. The strong Nullstellensatz reconciles this: sqrt((x^2)) = (x), so geometry sees only the radical.
Two ideals, one vanishing set, reconciled by the radical.
The dictionary it sets up — radical ideals to algebraic sets, primes to irreducible sets, maximal ideals to points — is the starting point of classical algebraic geometry and a motivation for the spectrum of a ring.