A reed gap is the gap between the reed and the slot in the reed plate (see the picture above). The gap height greatly influences how the reed plays. A wider gap requires and allows more pressure to make the reed sound. If you attack notes hard, a relatively wide gap can help keep the reed from "choking" and refusing to sound. A narrower gap allows less air to activate the reed. If you play softly, a relatively small gap can help the reed activate with a soft attack. If the gap is too small, like with the reed tip inside the slot, the reed will refuse to play.
The reed gaps need to be wider for longer reeds. In other words, the low notes should have slightly more gap between the reed and the reed plate than the high notes. In order for the harmonica to play smoothly and uniformly, the gaps must be consistent for every reed, with the slight adjustments applied for different length reeds. The nominal adjustment is for the gap to be about the same as the thickness of the reed.
For overblows, the reeds should be gapped relatively close to the reed plate. This can be crucial for obtaining overblows! An improperly gapped reed will just refuse to overblow.
Every bit of the reed should be above the reed plate, and the distance between the reed and the slot in the reed plate (the gap) should continually increase from the base of the reed (nearest the rivet) to the tip. If any of the reed dips into the slot, or if the reed arches up and then back down at the tip, the reed will not respond properly.
Harmonica reeds are essentially just brass springs that vibrate through slots in the reed plates to chop the air stream, which produces the sound. To adjust the reed gaps, just use your fingernails or a small tool to gently press the reed down, to close the gap, or up to increase the gap. After an adjustment is made, flick the tip of the reed a few times to get the reed to settle to its rest position--if you don't you can get fooled by the reed position. It can look one way, but revert back to where it was after you play a little--remember, it's a spring. Flicking the reed tip a few times is a good way to get the reed to settle so you can correctly determine its gap. It's best to bend the reed in very small increments, and not make over adjustments. Slight over adjustments are inevitable, but repeated bending one way, then the other, will weaken the reed and could even cause it to break.
By the way, when you go to increase the gap you may want to slide something thin under the reed tip to get hold of the reed. Be careful not to slide anything too far back toward the rivet. If you lift the base of the reed out of the slot you'll probably end up making the reed pitch flat. It is always a good idea to make sure your harps are in tune, and after gapping is a good time to check since you've got the harp open anyway.
Gapping is easy, safe, and a basic requirement for making a harp play well. Factory reeds are set to some average beginner gap, and are usually too wide--and most often inconsistent across the harp. I strongly recommend re-gapping your harps according to your personal playing style and needs.