This abelian group, which can also be described as the quotient of the direct product $\prod_{\mathbb N}\mathbb Z$ by the direct sum $\sum_{\mathbb N}\mathbb Z$, is isomorphic to the direct sum of the following two pieces. The first piece is a torsion-free, divisible abelian group (so you can view it as a vector space over $\mathbb Q$) of rank (i.e., dimension over $\mathbb Q$) equal to the cardinality of the continuum. The second piece is the direct product, over all primes $p$, of the direct product groups $\prod_{\mathbb N}\mathbb Z_p$, where $\mathbb Z_p$ is the additive group of $p$-adic integers. I believe this result is due to Balcerzyk.
Edit: I believe the Balcerzyk reference is "On factor groups of some subgroups of a complete direct sum of infinite cyclic groups" [Bull. Acad. Polon. Sci., Sér. Sci. Math., Astron., Phys. 7 (1959) 141-142.