Follow-up to Chandan Singh Dalawat's excellent answer: smooth curves over $\mathbb{F}_p$ can always be lifted to over $\mathbb{Z}_p$. But there are smooth projective surfaces over $\mathbb{Z}/p^n$ that cannot be lifted to $\mathbb{Z}_p$, see http://front.math.ucdavis.edu/0411.5469 .

Follow-up to Chandan Singh Dalawat's excellent answer: smooth curves over $\mathbb{F}_p$ can always be lifted to over $\mathbb{Z}_p$. But there are smooth projective surfaces over $\mathbb{Z}/p^n$ that cannot be lifted to $\mathbb{Z}_p$, see Link .