There are different ways to think about reciprocal space. For one, you can think of it as a purely geometrical thing. If you have a lattice with vectors [tex]\mathbf{a}_i[/tex], then the reciprocal lattice is defined by
[tex]\mathbf{a}_i \cdot \mathbf{g}_j = 2\pi\delta_{ij}[/tex].
It's a purely geometrical thing, and the above equation defines a lattice with vectors [tex]\mathbf{g}_i[/tex]. In mathematics this would probably be called a dual lattice, but in physics we call it a reciprocal lattice because of the units (g has units of 1 / length).
Another way reciprocal space shows up is if you look at the solution to the Schroedinger equation for a periodic potential, Bloch's theorem says that the wavefunctions are a product of two periodic functions, and of the form
[tex]\psi_{kn}(r) = u_{kn}(r) e^{i k \cdot r}[/tex]
where k is the socalled pseudomomentum vector, which serves as a quantum number. k is restricted to the first Brillouin zone, where [tex]k = l_1 g_1 + l_2 g_2 + l_3 g_3[/tex] where the l's are restricted to the range [0.5,0.5]. So k is restricted to wavelengths which are longer than a lattice vector. The other function [tex]u_{kn}(r)[/tex] is periodic within the unit cell, so if you expand it in planewaves, all the planewaves would be like [tex]n_1 g_1 + n_2 g_2 + n_3 g_3[/tex] where the n's are integers, thus these wavelengths are all the lattice constants divided by integers.
