This is an ambitious request that would take many posts to address properly.  The basic idea is that the grid cells in our forecast models are fairly coarse - typically with horizontal scales on the order of tens of miles.  Therefore, these models can only use basic physical equations to resolve processes that are larger than the grid cells.  Of course, lots of weather occurs on spatial scales that are much smaller than tens of miles!  We rely on parameterizations to account for all of these unresolved processes.

I will give a few common examples of phenomena that require parameterization.  Convective clouds (e.g., thunderstorms) have a big role in our weather and climate (we've definitely seen this theme in the ENSO Blog!).  These clouds typically occur on time and space scales much smaller than the time steps and grid scales in our forecast models, so we need ways to parameterize when these clouds develop and what their effects on the larger (the resolved) scales are.  On even smaller scales, clouds involve the formation of tiny cloud droplets that grow, shrink, and change phase, so we need cloud microphysics parameterizations to handle them.  Atmospheric turbulence encompasses a cascade of energy from large scales to small scales, and we need turbulence parameterizations for the smaller scale motions that we cannot resolve.  Gravity waves induced by flow over mountains are another example of phenomena not properly resolved by our forecast models, and we gravity-wave drag parameterizations to account for the effect of the gravity wave breaking high in the atmosphere. 

The fundamental assumption for parameterization is that we can account for the effects of unresolved processes in terms of processes that we can resolve.  For example, we may not be able to resolve individual thunderstorms, but we know the large-scale conditions that lead to their formation, and we have a good idea of their effects on the large-scale environment after they occur.  That doesn't mean that we can't do better at representing their effects!  As computer power grows and forecast models become finer and finer in scale, some processes that are currently parameterized will not need to be.