There is a very nice answer for abelian varieties.

Theorem 1.1 of Conrad and Brinon's [notes on p-adic hodge theory][1] gives a nice theorem in the case of Abelian varieties: let A be an abelian variety (for simplicity over Q), l a prime, and $p \neq l$ a second prime. Then A has good reduction (e.g. there exists a smooth model) iff the l-adic Tate module (equivalently the l-adic etale cohomology) is unramified at p. For A an elliptic curve an elementary proof of this is in Silverman's *Arithmetic of Elliptic Curves*, Theorem 7.1 (Criterion of Neron–Ogg–Shafarevich). 


The case l = p is in a sense the beginning of p-adic hodge theory. Grothendieck gave a nice criterion in terms of p-divisible groups: A extends to a smooth model iff each torsion subscheme A[n] extends to an integral model (in a compatible way). This is the same as saying the p-divisible group associated to A extends to an integral model. This is all explained very well in Conrad and Brinon's notes (Section 7). 

Later it was proved that this is equivalent to the p-adic Tate module being Crystalline. This is also in Brinon and Conrad's notes. 


Finally, this type of theorem fails for surfaces, even when $l \neq p$! I think Shenghao Sun knows an example where the l-adic etale cohomology is unramified at p, but the surface still had bad reduction.




  [1]: http://math.stanford.edu/~conrad/papers/notes.pdf