If $K$ is a number field (assume not a principal ideal domain), $O_k$ its ring of integers, we can ask which ideals $I$ of $K$ are principal. Let $P$ be a non-principal ideal in $K$. We can ask for which ideals $Q$ is $PQ$ a principal ideal. What is this situation referred to?
Second, If $I$ is a principal ideal of norm $p$, we have an element $e$ in $K$ of norm $p$. Similarly, consider the ideals $Q$ such that $PQ$ is principal ($P$ is a non-principal ideal). Hence $Q$ is an ideal of norm $q$, and we must also have another element $E$ in $K$ of norm $q$. How are the elements of norm $q$ constructed in this case?
The nonzero ideals in a Dedekind domain, such as $O_K$ fall into ideal classes. Two ideals $I$ and $J$ are in the same class iff $J=xI$ with $x\in I$. The principal ideals form one ideal class. The collection of ideal classes form a group under multiplication.
Let $P$ be non-principal. Then $PQ$ is principal iff Q is in the ideal class which is inverse to the ideal class of $P$: in symbols $Q\in[P]^{-1}$.
I'm not sure about your second question. I see no reason why the norm of the non-principal ideal $Q$ should be the norm of a field element.