A very well-known condition is that the Hilbert scheme of a smooth surface is smooth(I believe also . As David pointed out below, the Hilbert scheme of a smooth curve is smooth)smooth and equal to the symmetric product (since k[t] has only one finite dimension quotient of each dimension).
I don't know of any other examples, but one of the versions of Murphy's Law in algebraic geometry is roughly "if you don't have a good reason for a Hilbert scheme to not be horrible, it will be as horrible as you can possibly imagine."
