The ring of integers $\mathcal{O}_K$ of a number field $K$ of degree $n=[K:\mathbb{Q}]$ is a commutative integral domain whose underlying additive group $(\mathcal{O}_K,+)\simeq \mathbb{Z}^{n}$. So what can one say about the collection of all number fields $K$ of a fixed degree $n$? The answer is: nothing much at all if $n>5$, and a whole lot when $n\leq 5$.
We classify all commutative ring structures on the additive group $\mathbb{Z}^n$ for $n\leq 5$ (which is due to Bhargava and others). For instance commutative rings $R\simeq \mathbb{Z}^3$ are in bijection with isomorphism classes of binary integral cubic forms $Q(x,y)=ax^3+bx^2y+cxy^2+dy^3$. Where do such descriptions come from for the moduli of rings of rank $n$? Come and find out!
You need to know some ring theory to understand this talk, I can't think of any other prerequisites.