The best explicit criterion that I know is that of criterion of Kummer-Dedekind, which involves writing $K = \mathbb{Q}[t]/(P(t))$ and factoring $P(t)$ modulo the prime $p$. Then the factorization of $(p)$ in $\mathbb{Z}_K$ "has the same shape" as the factorization of $P(t)$ in $(\mathbb{Z}/p\mathbb{Z})[t]$: see e.g.
http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/dedekindf.pdf
This criterion does not apply to primes dividing the discriminant of the polynomial $P(t)$ but not the number field $K$.
At a more theoretical level, the Chebotarev density theorem gives some powerful asymptotic information: for instance, it says that the density of the set of primes which split completely in $K$ is $\frac{1}{[M:K]}$, where $M$ is the Galois closure of $K/\mathbb{Q}$. Also class field theory has things to say in the special case when $K/\mathbb{Q}$ is abelian.
In some sense, the general problem is unsolved: it is one of the things that we imagine we might know better if we knew a "non-abelian class field theory".