An ideal $I$ in the ring of integers $\mathcal O$ of some number field factors into a product $I = P_1^{e_1} \cdots P_r^{e_r}$ of prime ideals. Hence,
$$ \mathcal O/I \cong \prod \mathcal O/P_i^{e_i}. $$
So the question becomes: What is the structure of $\mathcal O/P^e$ where $P$ is a prime ideal?

The additive and multiplicative structures of these rings are described in the following survey papers.

> Elia, Interlando, Rosenbaum, [On the structure of residue rings of prime ideals in algebraic number fields Part I: unramified primes.][1] *Int. Math. Forum* 5 (2010), no. 53-56, 2795–2808. 
>
> Elia, Interlando, Rosenbaum, [On the structure of residue rings of prime ideals in algebraic number fields—Part II: ramified primes.][2] *Int. Math. Forum* 6 (2011), no. 9-12, 565–589.

[1]:http://www.m-hikari.com/imf-2010/53-56-2010/eliaIMF53-56-2010.pdf
[2]:http://www.m-hikari.com/imf-2011/9-12-2011/eliaIMF9-12-2011.pdf