Various types of heat transfer are good examples. A time-dependent temperature field $T(x,y,z,t)$ satisfies
$$ {\partial T \over\partial t}=\nabla \cdot (\alpha\nabla T), $$
where $\alpha=\lambda/(\rho C)$ is the thermal diffusion coefficient for thermal conductivity $\lambda$, specific heat $C$, and density $\rho$. The diffusion coefficient may vary over space. Or in one dimension,
$$ {\partial T \over\partial t} = \frac{\partial}{\partial x}\left(\alpha\frac{\partial T}{\partial x}\right).$$
Note that setting the left-hand side to zero will turn it into a Laplace equation (steady-state temperature).
A type I (Dirichlet) boundary condition would be to set the temperature to a fixed value at the boundary. Example: an object immersed in an ice-water mixture.
A type II (Neumann) boundary condition would be to set the heat flux to a fixed value. Heat flux is $q''=-\lambda \partial T/\partial x$. For example: a resistive heating element that dumps a fixed wattage of power into the system.
A type III (Robin) boundary condition could be radiative cooling: the heat flux at the boundary is proportional to the temperature difference with the environment. Suppose that the environment has temperature $T_0$; the heat flux condition at the rightmost boundary is then $\lambda\partial T/\partial x=h(T_0-T)$, where $h$ is a proportionality constant.
For wave equations, consider a standing wave in a pipe, with the displacement of the air molecules as your $u$ parameter. The three types correspond to a closed pipe end ($u=0$), an open pipe end ($u_x=0$), and a partially open pipe end.
