Technically, an “assault rifle” is a rifle that can work automatically (like a machine gun) as well as semi-automatically (one shot per pull of the trigger). Machine guns are illegal in most states and require an extensive background investigation (similar to getting a Secret clearance) and payment of a Federal “transfer tax”. Even though Federal law allows ownership of machine guns, you still have to live in a state that allows them.
There are two types of semi-automatic (and automatic) actions. One is the “recoil operated” (AKA “blowback”) action in which the force of the rapidly-burning powder (gunpowder burns rapidly, rather than explodes. A fine distinction, but there you are.) overcomes the mass and inertia of the bolt. The action is designed to not push back the bolt until the bullet is safely on its way down the barrel.
The second type is the “gas operated” action. Gasses that propell the bullet are taken from a port somewhere in the barrel. Most people know what an M-16 looks like from seeing them in the movies. The triangular piece near the muzzle (the front of the barrel) holds the front sight and is also the place where the gasses are ported from the barrel. There is a tube that runs from the front sight (in the M-16/AR-15) back to the bolt carrier. The gasses travel down the tube and act against the bolt carrier to push it back, cocking the hammer. A spring (in the M-16/AR-15 it is in the butstock) returns the bolt carrier assembly to the closed position, chambering a new round along the way. Some gas operated systems (as in the AR-18/Ar-180) use a piston. The gasses push a piston that pushes a rod that pushes the bolt carrier.
Although RickJay didn’t imply it, I’ll say it anyway: All “semi-automatic rifles” are not “assault rifles” (especially considering the technical description). There are semi-automatic actions in pistols, hunting rifles, shotguns, and even your little .22 “plinker”. This is one reason “gun nuts” oppose regulation of “semi-automatics”. (There are other reasons, but you can read all about them in GD.) An AR-15 so-called “assalt rifle” and a Ruger Mini-14 ranch/hunting/sporting rifle are functionally the same. They both fire the same round, they’re both gas-operated, and they’re both fed from detachable magazines. The difference is that one os black and the other comes with a wooden stock.
Semi-automatic pistols may have a sliding bolt (like the Luger P.08 or the Ruger MkII) or the whole top of the gun may move back (the “slide”, as in the Colt 1911 or the Beretta). Revolvers have a cylinder that holds the rounds. These are the “six-shooters” of the Old West, and the .38s you hear about in the old cop shows. There are also bolt-action pistols that are used for target shooting and hunting.
Bolt-action means that there is a bolt that must be manually operated. If you saw Saving Private Ryan, the sniper used a bolt-action M1903 rifle. The German Gewehr 98s in the movie were also bolt-action. Bolt-action rifles are very popular with hunters, and have also been used on some shotguns.
Lever-action rifles have a lever below them. Think of The Rifleman series or any of the old westerns. Manipluating the lever opened the action, ejected the spent case, cocked the hammer and chambered a new round. Interestingly, one of the first “machine guns” was made from a lever-action rifle. A “technology demonstrator”, it had a flap over the muzzle that was attached by a rod to the action. When the round was fired the gasses would move the lever out of the way of the bullet and work the action at the same time.
Pump-action firearms have a slider under the barrel. These are popular on shotguns (“Rack-rack!!!”) and have also shown up on hunting rifles and “arcade guns” (.22 rifles used in shooting galleries).
Break-actions are usually on shotguns. There is a pivot on the receiver that allow the gun to be “broken”. The shells are ejected or manually extracted and new rounds are inserted. Closing the action cocks the gun and makes it ready to fire. There have also been break-action pistols and rifles.
Falling-blocks were used in early cartridge guns, for example the Sharps 1873 and the earlier Sharps non-cartridge breechloader, and the Martini rifles featured in Zulu. A lever dropped the breechblock and allowed a round to be inserted or a spent case to be extracted. Most had a hammer that needed to be cocked as well.
Single-action and double-action are subtypes. A single-action firearm requires that the hammer be cocked manually for each shot. A double-action cocks the hammer when the trigger is pulled. A single-action automatic requires the hammer to be cocked manually (either with the thumb or when the slide is operated) for the first round. The hammer is cocked automatically for subsequent rounds. A double-action automatic cocks the hammer when the trigger is pulled for the first round, but the hammer is cocked automatically for the rest of the shots. Revolvers operate in the same way, except that the hammer is not cocked automatically. Most double-action revolvers can also be used single-action. Since it takes such a hard trigger pull to cock the hammer on a double-action revolver, accuracy may suffer.
A “hair trigger” is a trigger assembly that has been modified to release the hammer at the slightest touch of the trigger. This is useful for competition shooting, but is dangerous on a firearm that is to be carried. Some older rifles had two triggers. One would release most of the pressure on the trigger assembly and the other would be used to fire the round. The latter would require a much lighter pull than would be required if the first trigger was not pulled.