the reciprocal lattice
The reciprocal lattice is the crystal's shadow in wavevector space, its Fourier partner. If the real (direct) lattice tells you where the atoms repeat in position, the reciprocal lattice tells you which waves fit the crystal perfectly, which spatial frequencies the crystal is built from. It lives not in ordinary space but in k-space, the space of wavevectors, with units of inverse length.
Formally, the reciprocal lattice is the set of all vectors G such that exp(i G . R) = 1 for every direct lattice vector R. It is built from primitive reciprocal vectors b1 = 2 pi (a2 x a3)/[a1 . (a2 x a3)], and cyclic permutations, which satisfy the duality relation b_i . a_j = 2 pi delta_ij. Then G = m1 b1 + m2 b2 + m3 b3 for integers m_i, and the reciprocal lattice is itself a Bravais lattice. The famous consequence is diffraction: X-rays or electrons scatter constructively from a crystal exactly when the scattering wavevector change equals a reciprocal lattice vector G, the Laue condition, which is why a diffraction pattern is a direct photograph of the reciprocal lattice.
Everything periodic in a crystal, the potential, the charge density, is naturally expanded in these G vectors, so the reciprocal lattice is the home coordinate system of band structure and phonon dispersion. A caution on conventions: some texts omit the factor of 2 pi (folding it into the plane-wave definition instead), so always check whether b_i . a_j equals 2 pi or 1 in a given book.
A cubic direct lattice of spacing a has a cubic reciprocal lattice of spacing 2 pi / a; a bigger crystal cell means a finer, more closely spaced reciprocal lattice.
Long in real space maps to short in reciprocal space; the two are inversely scaled.
The reciprocal lattice is not just a bookkeeping trick: diffraction peaks literally sit at its points, so an X-ray pattern is a map of the reciprocal lattice, not the real one.