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The reciprocal lattice is the Fourier dual of the direct (real-space) Bravais lattice. Its lattice vectors b_i are defined by a_i · b_j = 2π δ_{ij}, so every reciprocal lattice vector G satisfies eiG·R = 1 for all direct lattice vectors R. The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice — the region of k-space closer to the origin than to any other reciprocal lattice point. It is the fundamental domain for describing wave phenomena in crystals: electron band structures, phonon dispersions, and diffraction conditions are all naturally expressed within the first Brillouin zone.
If the direct lattice tells you where the atoms are, the reciprocal lattice tells you how waves interact with those atoms. The reciprocal lattice is constructed by defining vectors b_1, b_2, b_3 that satisfy a_i · b_j = 2pi delta_{ij} with respect to the primitive direct lattice vectors. In three dimensions, the explicit formula is b_1 = 2pi (a_2 x a_3) / [a_1 · (a_2 x a_3)], and cyclically for b_2 and b_3. Any vector G = h b_1 + k b_2 + l b_3 (with h, k, l integers) is a reciprocal lattice vector, and it has the fundamental property that eiG·R = 1 for every direct lattice vector R.
This property makes the reciprocal lattice the natural setting for Fourier analysis of periodic functions. Any function with the periodicity of the crystal — the electron density, the potential, the charge distribution — can be expanded in a Fourier series whose wavevectors are exactly the reciprocal lattice vectors. Diffraction experiments (X-ray, electron, or neutron) directly measure the Fourier components of the electron density, which is why diffraction patterns are essentially images of the reciprocal lattice. The von Laue condition for constructive diffraction is Delta k = G: the change in wavevector must equal a reciprocal lattice vector.
The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice — the set of all points in k-space that are closer to the origin than to any other reciprocal lattice point. It is constructed by drawing perpendicular bisecting planes between the origin and each neighboring reciprocal lattice point, then taking the innermost volume. For a square lattice the first Brillouin zone is a square; for FCC it is a truncated octahedron; for BCC it is a rhombic dodecahedron. The FCC and BCC reciprocal lattices are duals of each other, leading to the elegant result that the Brillouin zone of one has the shape of the Wigner-Seitz cell of the other.
The Brillouin zone matters because Bloch's theorem tells us that electronic states in a periodic potential are labeled by a wavevector k, and states differing by a reciprocal lattice vector G are physically identical. The first Brillouin zone contains one representative of every distinct k, making it the minimal domain needed to describe all electronic (or phonon) states. High-symmetry points and paths within the Brillouin zone — labeled Gamma, X, L, K, M, and so on depending on the lattice — are where band structures are conventionally plotted and where important physical phenomena (band crossings, van Hove singularities, Fermi surface features) tend to occur.