Are there different classifications or just the one? How does our galaxy group compare with others?
A galaxy group is a set of galaxies that are close together and gravitationally bound, i.e. barring outside influences they will stay together indefinitely. That means a galaxy that is in the space occupied by the group but "passing through" it at a high velocity would not be considered part of the group.
The upper limit for calling it a group is roughly 50, bigger accumulations of galaxies are called clusters (which may consist of many groups). The borders between adjacent groups may be uncertain.
These aren't mathematical definitions and actual usage is often based on tradition more than on measurements. It's just like geography: you could debate endlessly whether a given piece of high ground should be called a hill or a mountain, and whether something is a separate mountain or a subpeak, but sensible people recognize it as a waste of time, make up a clear but arbitrary numeric distinction without strongly insisting on its usage.