I'm new to group theory and am having trouble grasping the concept of cosets. For $N < G$, what exactly is $G/N$?

As an example consider the [quaternion group](http://en.wikipedia.org/wiki/Quaternion_group) $Q$ and it's center $Z = \{1, -1\}$. Then the left and right cosets are the same: $\{1, -1\}, \{i, -i\}, \{j, -j\}, \{k, -k\}$.

So cosets aren't groups, but is the collection of these cosets a group, and for what operation? (I'm pretty sure that it is in this case, because I read it's isomorphic with $V\_4$, but I don't understand if it is general and how it works).