The Borsuk-Ulam theorem states that for every continuous map f : S^n -> R^n, there exists a point x in S^n such that f(x) = f(-x): some pair of antipodal points must map to the same value. Equivalently, there is no continuous map S^n -> S^{n-1} that commutes with the antipodal map. This theorem has striking consequences: it implies the ham sandwich theorem (any n measurable sets in R^n can be simultaneously bisected by a single hyperplane) and that at any moment, there exist two antipodal points on Earth with identical temperature and pressure.
The Borsuk-Ulam theorem is one of the most elegant and applicable results in algebraic topology. The antipodal coincidence version states: for any continuous map f : S^n -> R^n, there exists a point x with f(x) = f(-x). In the n = 2 case, this has the beautiful meteorological interpretation: at any moment, there exist two antipodal points on the Earth's surface with identical temperature and atmospheric pressure (modeling temperature and pressure as continuous functions from S^2 to R^2).
The proof for general n uses the theory of the antipodal action on spheres and projective spaces. The Z/2Z-action on S^n given by x -> -x is free (no fixed points), and the quotient is real projective space RP^n = S^n / (x ~ -x). A continuous equivariant map g : S^n -> S^{n-1} (satisfying g(-x) = -g(x)) would descend to a continuous map g-bar : RP^n -> RP^{n-1} on the quotients. The key topological input is about the cohomology (or homology with Z/2Z coefficients) of projective spaces: H^k(RP^n; Z/2Z) = Z/2Z for 0 <= k <= n, and the generator alpha in H^1 satisfies alpha^n != 0 in H^n. The induced map g-bar^* would have to be an isomorphism on H^1 (by the equivariance condition), hence send alpha to alpha, and therefore send alpha^n != 0 to alpha^n. But alpha^n in H^n(RP^{n-1}; Z/2Z) = 0 (since RP^{n-1} has no cohomology in degree n). Contradiction.
The ham sandwich theorem is the most famous application. Given n measurable sets (or "ingredients") in R^n (think: bread, ham, cheese in R^3 for n = 3), there exists a single hyperplane that simultaneously bisects all n sets into equal-volume halves. The proof parametrizes hyperplanes by their normal direction on S^{n-1} and uses Borsuk-Ulam to find a direction where all n bisecting offsets agree. This result is used in computational geometry (fair division algorithms) and in measure theory.
Another important consequence is that S^n does not embed in R^n: any continuous map S^n -> R^n must send some pair of antipodal points to the same image, so no such map can be injective. This is a stronger statement than the invariance of dimension (which says R^n is not homeomorphic to R^m for n != m) and provides a clean topological obstruction to "dimensionally reducing" spheres.
The Borsuk-Ulam theorem also has discrete combinatorial analogues. The necklace splitting problem (splitting a necklace with n types of beads fairly between two people using at most n cuts) follows from a discrete version of Borsuk-Ulam. Tucker's lemma (a combinatorial analogue on triangulated spheres) is equivalent to the Borsuk-Ulam theorem and is used in fair division algorithms and computational complexity (the PPAD complexity class). These connections between continuous topology and discrete mathematics demonstrate the unexpected reach of the Borsuk-Ulam theorem beyond its original setting.