I have always wondered and once I even got it, but then completely forgot. I understand that gravity causes high and low tides in oceans, but why does it occur on the other side of Earth?
Imagine that we have a very massive object in space.  At some distance away (call it ten units) we release three tennis balls in a row:



The tennis balls all fall towards the massive object.  But because gravity goes like distance squared, the nearer balls feel a stronger attraction than the farther balls, and they move apart from each other:



You're riding on the middle tennis ball. You feel like you're in free fall, in a good inertial frame. You look towards the heavy object and you see the leading tennis ball moving away from you. You look away from the heavy object and you see the following tennis ball moving away from you. The heavy object is pulling the three tennis balls apart.

Likewise, if you had three objects at the same distance falling towards the massive object,



you'd see them converge as they all fell along slightly different rays towards the same center. This gives the tidal compression. You can imagine the process of launching a whole constellation of tennis balls, choosing the center one as your "rest frame," and having their motions approximate the arrow pattern in Joshua's figure.

The situation stays essentially the same if you add angular momentum, except that then your tennis ball constellation doesn't crash onto the massive object.
