According to the Big Bang theory (a simplified version of it, anyway), the entire universe started out in an incredibly dense "singularity," then exploded, with matter spreading out to become what we now know as stars, planets and galaxies. Since all the galaxies started at the same point and the universe is continually expanding, the galaxies should always be moving farther apart. The common comparison is with raisins in a loaf of baking bread, or dots on a balloon that is being inflated. Movement is always outward, away from the centre, so there's no way the raisins or dots can run into each other.

However, we do know that galaxies have collided, and are colliding. The usual explanation I've seen for this is that the force of gravity — galaxy attracting galaxy — can overcome the force of space-time expansion.

Galaxies are moving away from the centre of the universe (where the universe was "born"), but they also move about within their own clusters because of intergalactic gravity. In general terms, these galaxies are moving away from the centre of the universe, but in local terms they're orbiting each other or drifting toward each other, and sometimes they get close enough to collide.

See: Discover Magazine, Bad Astronomy and Universe Today Forum, Curious About Astronomy?