Non-Draining Drippers do not let the system drain out once the water supply has been turned off. This helps you give exact amounts of water to every plant. Normal drippers will allow water to drain out at the lowest point in the system giving more water to either the lowest plants or the last plants on a line. This becomes even more pronounced if you have a green house with irrigation lines run above the plants, all that water in the lines will drain out if not stopped. These style drippers will do just that. Upon the water being turned off they will close up once the pressure within the system lowers to a certian point and they will be rated to hold back a certian factor of water column.
Check Valve drippers sort of do a little of the non-draining part but with a difference. They are more focused on not letting anything back "IN" to the dripper. If you have a system and the drippers may be laying in the dirt or even in a wet area when the water source is turned off the water will want to drain out and it will go to the lowest dripper. While doing this it creates a vacumn in the line which will try to be relived by sucking air, ware or dirt back in through dripper above where water is still draining out from. These dripper do not allow this to happen thus not getting dirt back into the dripper but not back into the line to clog other drippers.
Schools out, go buy some stuff! :-)