Janus said:
It depends on how close the galaxies are. Galaxies in our local group are close enough for there mutual gravitational attraction to hold them together against the expansion of the universe.
Galaxy clusters, such as ours are spread out far enough from each other that their gravity isn't enough to keep them together, so they recede from each other.
That's a part of it. The other important part is initial conditions. It's all about velocity.
To argue by way of analogy, one rule that we seem to more or less agree upon for our daily lives is that if we throw something up, that something is always going to come down. Well, since Newton we've known that this isn't strictly true: if we throw something up fast enough, it won't come down at all. The velocity at which this happens is known as the "escape velocity". Basically, an object at escape velocity is traveling so fast that even though the gravitational pull is reducing its velocity, it's traveling so fast that the pull can't reduce its velocity enough to ever stop the object.
This can be thought of in multiple ways. Perhaps the easiest way to compute it is to just use the gravitational potential energy:
U = \frac{-Gm_1 m_2}{r}
Where G is Newton's constant, m_1 and m_2 are the masses of the two objects, and r is the distance from that object's center. If we want to escape this object, all we need to do is to have more kinetic energy than potential energy. That is:
\frac{1}{2} m_1 v^2 > \frac{-Gm_1 m_2}{r}
If this is true, then the object will simply escape.
Now, knowing this fact, it shouldn't be too hard to understand that if we have objects that are moving fast enough away from one another compared to how far away they are, then they will continue to do so.
