Listing all the albelian groups of order 900

The systematic approach is to write down the factorisation $ 900 = 2^23^25^2 $. Then, for each prime factor, find all partitions of its exponent. Combine all these partitions to get the following list (I write $C_n$ instead of $\mathbb{Z}_n$ because I'm lazy):

$\begin{array}c C_4 \times C_9 \times C_{25} ( \cong C_{900} ) ,\\ C_2 \times C_2 \times C_9 \times C_{25} ,\\ C_4 \times C_3 \times C_3 \times C_{25} ,\\ C_4 \times C_9 \times C_{5} \times C_5 ,\\ C_2 \times C_2 \times C_3 \times C_3 \times C_{25} ,\\ C_2 \times C_2 \times C_9 \times C_{5} \times C_5 ,\\ C_4 \times C_3 \times C_3 \times C_{5} \times C_5 ,\\ C_2 \times C_2 \times C_3 \times C_3 \times C_{5} \times C_5. \end{array}$

The structure theorem for finitely generated abelian groups, also known as fundamental theorem, confirms that these are indeed all abelian groups of order $900$ and that they are pairwise non-isomorphic.

Nothing wrong with the answer by m_l. The other way to state the structure theorem is the one in Phira's comment. This gives you $C_{900}$, $C_2\times C_{450}$, $C_3\times C_{300}$, $C_5\times C_{180}$, $C_6\times C_{150}$, $C_{10}\times C_{90}$, $C_{15}\times C_{60}$, and $C_{30}\times C_{30}$.