When are Hilbert schemes smooth? I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a Hilbert scheme to be smooth?
 A: For some more examples of smooth HS see A.P.Staal: The ubiquity of smooth Hilbert schemes, arxiv AG 31.Jan. 2017.
A: The Hilbert scheme of n points on a 3-fold is not smooth for n sufficiently large, but the exact value of sufficiently large is unknown. See the chapter on Hilbert schemes in "Combinatorial Commutative Algebra", by Miller and Sturmfels.
A: A very well-known condition is that the Hilbert scheme of a smooth surface is smooth.  As David pointed out below, the Hilbert scheme of a smooth curve is 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."
A: I don't know of many global conditions for a Hilbert scheme to be smooth/singular.  Ben's answer probably gives the most interesting example of a smooth Hilbert scheme, namely the Hilbert scheme of n points on a smooth surface.
Here are two more examples of smooth Hilbert schemes.
1)  The Hilbert scheme of hypersurfaces of degree d in PP^n.  Such hypersurfaces are parametrized by homogeneous degree d polynomials in n+1 variables, and hence this Hilbert scheme is a projective space of dimension n+d choose d.
2)  The Hilbert scheme of linear subpsace of dimension d of PP^n.  This is just the Grassmanian Gr(d+1,n+1).
A: Here is yet another example of a smooth Hilbert scheme.  Let $X$ be a smooth degree 3 hypersurface in projective space of dimension $n \geq 3$ (say, over an algebraically closed field), and let $H$ be the Hilbert scheme of lines on $X$ (i.e., corresponding to Hilbert polynomial $t + 1$).  
The tangent space to $H$ at a point $[L]$ (corresponding to a line $L$ in $X$) is $H^0(L, N)$ where $N$ is the normal bundle of $L$ in $X$.  The rank of $N$ is $n - 2$ and the degree of $N$ is $2n - 6$ (you can see this by looking at the standard tangent bundle and normal bundle sequences).  Every vector bundle on $L = \mathbb{P}^1$ splits into the direct sum of line bundles.  Then the degree of each rank 1 summand of $N$ is at most 1 ($N$ injects into the normal bundle of $L$ in $\mathbb P^n$) and then you can show that no piece can have degree less than $-1$.  This allows us to conclude that $H^1(L, N) = 0$.  This means that $H$ is smooth at the point $[L]$ (see for example Kollár's book Rational Curves on Algebraic Varieties, Chapter 1, where he explains the infinitesimal behavior of the Hilbert scheme).  Since this is true for any line $L$ in $X$, the Hilbert scheme is smooth.  
The same argument works for lines on a smooth Quadric.  In the same book, Kollár proves that for a general degree $d$ hypersurface $X$ in $\mathbb P^n$, the Hilbert scheme of lines on $X$ is smooth. 
