For an algebraic number field $K$ with ring of integers $_K$, the numerical norm $ℕ()$ of an ideal $⊆_K$ is defined to be the (finite) index of abelian groups $[_K:]$, or equivalently $|_K/|$.
Why is $_K/$ a useful thing to look at? Exactly what does $ℕ()$ tell you about $$? In particular, when and why is it true that $ℕ()=p$ means that $p_K=$ for some other ideal $$?
Don't focus only on the norm of an ideal in isolation, but also on the norm mapping on all ideals. Using norms of ideals in the integers of a number field $K$ leads to the zeta-function of $K$: $$ \zeta_K(s) = \sum_{\mathfrak a} \frac{1}{{\rm N}(\mathfrak a)^s} $$ when ${\rm Re}(s) > 1$. This is a pretty useful construction.
In modular arithmetic in $\mathbf Z$ we have the well-known formula $$ \varphi(m) = m\prod_{p \mid m} \left(1 - \frac{1}{p}\right), $$ where $\varphi(m) = |(\mathbf Z/m\mathbf Z)^\times|$. It turns out that there's an analogous formula for the units modulo a (nonzero) ideal in the integers of every number field: $$ |({\mathcal O_K}/\mathfrak a)^\times| = {\rm N}(\mathfrak a)\prod_{\mathfrak p\mid \mathfrak a}\left(1 - \frac{1}{{\rm N}(\mathfrak p)}\right). $$
Norms in various senses show up all over the place in number theory. It might as well be called "norm theory".