An open bolt firearm in which the act of closing ('dropping') the bolt to close the chamber results in the loaded cartridge being immediately fired. By definition, the chamber on such firearms does not lock closed, and is sealed by inertia and spring pressure only, so it is generally only appropriate with less powerful cartridges.
Guns are designed with an open bolt for two general reasons. First, they are mechanically simpler; rather than having to have locking mechanisms for the chamber, along with associated mechanisms for safe unlocking such as delayed blowback, recoil operated or gas operated systems, the only moving mechanical parts need be the loading mechanism and the trigger mechanism. Generally, open bolt guns do not even have a separate moving firing pin; instead, a fixed pin is mounted on the bolt face, and when the bolt closes that pin strikes the primer. This simplicity can result in a cheaper, lighter and/or more reliable gun.
The second primary reason for using an open bolt is related to the first: if you are building an automatic weapon, the simpler mechanism is greatly to be desired for purposes of reliability, speed of operation and heat management. Generally, when an open bolt gun is cocked, the bolt is held back from the chamber by the sear, and the trigger releases the bolt to slam forward and fire the gun.
This comes with a significant accuracy penalty, which is why semi-automatics are only rarely open bolt (but some do exist). The penalty is the result of two factors: moving mass and lock time. The first should be fairly self-evident: when firing the gun results in a significant mass moving within it, the gun will likely deviate slightly from the point of aim due to the shifting mass before the round is actually fired. Second, this system means that there is a much more significant delay (known as lock time) between the moment the trigger sear is released and the time the cartridge fires, so the gun has more time to move off-target before the round leaves the barrel. This further complicates the first problem. In a fully automatic firearm, of course, accuracy at this level is less important than speed and reliability of operation - more rounds downrange can to some degree compensate for lack of pinpoint accuracy.
There are safety and environmental considerations as well. An open bolt gun presents, in most cases, more avenues for foreign matter to enter the mechanism when it is being carried in 'ready' state, which is why many such have dust covers which are either manually or automatically opened before firing. The safety issue arises when such a gun is carried uncocked, or uncharged. If a cartridge is in the chamber, movement of the gun can cause the bolt to move slightly back and forth due to inertia, and unless the bolt is immobilized by a safety mechanism (which not all have) it is possible for such disturbance of the gun to move the bolt far enough for its return trip to strike the primer of the ready round and fire it without the trigger being pulled. The M3 "Grease Gun", of WWII fame, had a dual-purpose dust cover for just this reason - when it was closed, it also locked the bolt in place to prevent this behavior.
Fully automatic closed bolt guns do of course exist. The AR-15 family and derivatives and the Heckler & Koch MP5 are just two examples of such. These weapons are much more useful in semi-automatic operation than open bolt guns.