In SCUBA terms, a squeeze occurs during descent, while a reverse squeeze occurs during ascent. This involves the effects of increasing or decreasing pressure on the air spaces of the body and artificial air spaces like the mask. Some squeezes, like mask squeeze, can be equalized to prevent injury, others like tooth squeeze can't, so they might cause the diver to have to abort a dive. Bodily air spaces that can be affected include the lungs, inner ears, and intestines, and some artificial air spaces include the mask and an air pocket in a tooth from an improperly filled cavity.
The first thing to remember is that at the surface, your body is in equilibrium at 1 atmosphere of pressure. Every 33ft/10m you descend the pressure on the body increases by that same amount, so the pressure at 10m is 2 atmospheres, and 20m is 3 atmospheres, etc, and the volume of the air spaces that we're concerned with is the reciprocal of that number (Boyle's Law). So, at 20 meters, the pressure is three times greater, the volume of air is one third what it was on the surface. This can create a sucking, squeezing sensation in any air-filled cavity that isn't easily equalized. (The lungs are equalized because you're breathing compressed air at the same pressure as the water around you, though there is still a sensation of wearing something tight around the chest)
Ear squeeze can be avoided by equalizing the ears frequently during descent, using the Valsalva maneuver: pinching the nose and blowing gently against it. Blowing hard won't help the ears clear faster, and can actually damage them. Mask squeeze is that sucking feeling that your eyes are being pulled out of their sockets, but it is easily relieved by exhaling slightly through the nose into the mask. (Ever wonder why you can't wear goggles while SCUBA diving? That's why!) Tooth squeeze can't be easily relieved, so the dive may have to be aborted. You should go see a dentist if this happens so that they can check the cavity, and maybe refill it.
Reverse squeeze happens during the decreasing external pressure that is experienced during ascent. As the diver ascends, air that is trapped in the body expands and can cause discomfort. Air in the ears usually lets itself out gradually, though wiggling the jaw can help. Gas build up in the intestine can be very uncomfortable, and the ascent should be suitably slow to allow it to escape via the normal route. (All ascents should be slow to prevent decompression illness, so this should not be a problem.)
During descent, you have to put air in to equalize with the surroundings, and during ascent you have to let it out. In any case, if it's uncomfortable or hurts, stop and either float up a metre or so, or sink a little and wait until the pain goes away. depending on whether you're making a descent or ascent at the time.