Abstractly, it's the ring $\mathbb Z[u]$ in which $u^2=0$, and so its elements are of the form $a+bu$, with $a,b\in\mathbb Z$.
In this sense, it's like $\mathbb Z[i]$ in which $i^2=-1$. But they are very different rings, of course.
If you want to understand $\mathbb Z[u]$ a little better, you may want to find its units, its zero divisors, its nilpotent elements, its idempotents, its ideals, etc.
In particular, you'll find that there are no non-trivial idempotents in $\mathbb Z[u]$ and so it cannot be expressed as the product of two rings.