The terms describe the point from which the firearm is fired.
Closed Bolt: The bolt is closed on a chambered round. When you squeeze the trigger, a hammer (usually) is released to strike the firing pin, which ignites the cartridge. On a semi auto, the case is then extracted, a fresh cartridge is chambered and the bolt comes to rest, as before, on a chambered round. The AR15/M16, MP5, and a boatload of other rifles and pistols use this method.
Open Bolt: At the point of firing, the bolt is to the rear (open). When the trigger is squeezed, the bolt is released to strip a round from the magazine/belt and insert it into the chamber. As part of this same cycle, the hammer is then released to strike the firing pin and fire the round. The conclusion of the cycle is the extraction of the case and the return of the bolt rearward. A lot of belt-fed support weapons, like the M60, operate this way. This ensures that the chamber is always open prior to and after firing, so that it can cool (and so that a round cannot cook off under extreme heat). It's not the most accurate method, because the movement of the bolt can be significant.