nt.number theory - prime ideal factorization in an extension field

The best explicit criterion that I know is the 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 $(mathbb{Z}/pmathbb{Z})[t]$: 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".

Addendum: After copyediting your question, I interpret is as being especially interested in determining which primes $p$ remain prime in $K$ (or are $textbf{inert}$). Again the Chebotarev Density Theorem is helpful for this: suppose for simplicity that $K/mathbb{Q}$ is Galois. Then there exist primes $p$ which remain inert in $K$ iff the Galois group of $K/mathbb{Q}$ is cyclic, in which case the density of such primes is $frac{varphi([K:mathbb{Q}])}{[K:mathbb{Q}]}$. So for example, if $ell_1$ and $ell_2$ are distinct prime numbers and $K = mathbb{Q}(sqrt{ell_1}, sqrt{ell_2})$, the Galois group of $K/mathbb{Q}$ is isomorphic to $mathbb{Z}/2mathbb{Z} times mathbb{Z}/2mathbb{Z}$, so there are no inert primes.