Suppose that we have a lattice in $\mathbb{R}^n$, that is, a discrete additive subgroup of $(\mathbb{R}^n, +).$
We can define dual lattice $L^*$ as $$ L^* = \{x \in span(L) \; | \; \forall y \in L \; \;\langle x|y \rangle \in \mathbb{Z}\}$$
Using this definition we can derive some interesting properties of the original lattice $L,$ such as transference theorems and so on.
The definition is clear, but it is not obvious the motivation and intuition behind this object. Why do we want the scalar product to be in $\mathbb{Z}?$ How does it "look like"?
In physics of scattering from lattices (x-rays, electrons, neutrons), one gets bright spots corresponding to constructive interference of waves. The momentum transfer of the photons/electrons/neutrons is $\vec p$. The scattering from an atom at $\vec r$ is given by a term proportional to $e^{i\vec p\cdot\vec r}$. In the lattice, a different atom is at $\vec r+\vec R$, where $\vec R$ is a vector of the original lattice. So when one calculates the scattering intensity from all atoms in the lattice, you obtain a quantity proportional to $$\left|\sum_{\vec R}e^{i\vec p\cdot\vec R}\right|^2$$ It is easy to see that you have constructive interference where $\vec p\cdot\vec R=2\pi$ (or any multiple integer of $2\pi$). If we define the wavevector transfer as $\vec k=\frac{1}{2\pi}\vec p$, then the constructive interference condition is $\vec k\cdot\vec R=n\in \mathbb Z$. Therefore you get intensity when the wavevector transfer is a lattice vector in the reciprocal space of the position lattice.