A periodic crystal lattice is naturally described not only in real space but also in reciprocal space. If real-space primitive vectors are $\mathbf a_1,\mathbf a_2,\mathbf a_3$, reciprocal primitive vectors $\mathbf b_i$ are defined by $$\mathbf a_i\cdot\mathbf b_j=2\pi\delta_{ij}.$$ A reciprocal-lattice vector is $$\mathbf G=h\mathbf b_1+k\mathbf b_2+\ell\mathbf b_3,$$ with integers $h,k,\ell$.

Why introduce a second lattice? A function with crystal periodicity can be expanded in Fourier components whose allowed wavevectors differ by reciprocal-lattice vectors. Periodicity in position becomes discreteness in wavevector.

For a simple cubic lattice of spacing $a$, $$\mathbf a_1=a\hat x,\quad \mathbf a_2=a\hat y,\quad \mathbf a_3=a\hat z,$$ so $$\mathbf b_1=\frac{2\pi}{a}\hat x,$$ and similarly for $y$ and $z$. The reciprocal lattice is therefore also simple cubic, with spacing $2\pi/a$.

Planes in the real crystal, Fourier components of periodic potentials, diffraction conditions and electron crystal momentum all become simpler in reciprocal space. The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice and contains a nonredundant set of crystal wavevectors.

Reciprocal space is not a second physical crystal. It is the Fourier-dual geometry of translational periodicity, and it is the natural coordinate system for diffraction, phonons and electronic band structure.