Any nonsingular projective variety over $\mathbb{C}$ is easily seen to be a smooth manifold. Presumably the same is not true for algebraic varieties  one would not expect varieties with singular points to have a smooth structure. But do there exist nonsingular varieties that are not smooth manifolds?

Every nonsingular algebraic variety over $\mathbb C$ is a smooth manifold. See for instance: http://en.wikipedia.org/wiki/Manifold under "Generalizations of Manifolds". In fact, Arminius' suggested answer in the comments seems to give a proof of this fact, and I'll attempt to flesh it out a small amount. Every algebraic variety is locally a quasiaffine variety. So we may take an open cover $U_i$ of the variety, where each $U_i$ is a closed subset of an open subset of affine nspace. We may then check smoothness at each point of $U_i$ via the Jacobian criterion. The same procedure illustrates that each $U_i$ is a complex manifold. Since the gluing maps are algebraic, they are smooth, and hence our nonsingular variety is also a smooth manifold. 


Every nonsingular variety over $\mathbb{C}$ is a smooth manifold, period. Take any affine open cover $X=\cup U_i$. Then each $U_i$ is a smooth manifold, and the transition maps are algebraic, so in particular, smooth. Thus, manifold. 


If $k$ is a complete valued field (e.g. $\mathbb{Q}_p$, etc.), one may define analytic manifolds over $k$ in the natural way. Precisely, these are topological spaces that locally look like open balls in $k^n$ and the transition functions must be analytic. Then the $k$points of a smooth variety over $k$ is an analytic manifold (over $k$); Charlie's reasoning for the case $k = \mathbb{C}$ works for any $k$ as above. 

