An analytical solution is one that can be explicitly written down, also called a closed form solution. Analytical solutions can not be found for many partial differential and ordinary differential equations (most in fact). In the case when an analytical solution has not been found yet, numerical methods may be applied to get an approximate solution.
For instance a well known equation with an analytical solution is the wave equation. On the other hand, a well known equation that doesn't have an analytical solution is the Navier-Stokes equation.