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. 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. Int. Math. Forum 6 (2011), no. 9-12, 565–589.
