Bits have two main parts: the mouth piece and the shank/ring piece.
Snaffles lay calmly in the mouth until the reins are pulled. First, a very gentle pressure will just pull them against the lips. The hinge allows the rider to work each side of the mouth independently. With more pull, the bit will bend sharply in the center, pressing up on the roof of the mouth in the center and coming together at the rings. Used correctly, the pressure at the sides will come down on the bars and exert pressure there. If one side is pulled more than the other, pressure will first be exerted on the corner of the mouth and bar on that side; as more pull is exerted pressure will come against the far side of the mouth.
D rings and full cheek rings add to the lateral pressure of the additional pull in turning. I'll get to leverage in a sec.
All snaffles have that basic action. A French link adds a 2nd joint, which lessens the action on the roof of the mouth and eases the nutcracker effect on the sides of the mouth. A Dr. Bristol looks alot like the French link, but the little link is tilted so it doesn't lie flat but adds a bit of extra "bite" to the cuing.
An unjointed bit without a port can be straight or mullen (curved to match the tongue). These bits are not as subtle as the snaffle, as it is difficult to be so gentle as to independently cue one side of the mouth. Pull slightly to one side, and the horse will immediately feel it on both sides. THese bits put most of the pressure on the tongue and bars.
A ported bit has a numger of purposes. Most people think of it as a correction bit, as it can be used severely to apply active pressure on the roof of a horse's mouth. It is actually intended to provide extra room for the horse's tongue, and be a PASSIVE encouragement for good head set. If the horse has his head set correctly, the bit is shaped to his mouth and perfectly balanced to hang in the most comfortable position. Whether he raises or lowers his head, the weight of the shanks and reins keeps the port in the same position, so that he feels pressure on his tongue when his head goes up or on the roof of his mouth when his head goes down, which encourages him to carry himself correctly without the rider doing it.
These are used with shanked bits, whether the Kimberwick(Uxeter) or pelham or fullbridle bit for English or the many varieties of Western curb shanks.
One sacrifices the independent action of the snaffle to get the self carriage of the curb. It's a tradeoff.
When pressure is placed on the curb shank, the mouthpiece is the fulcrum of a lever betwen the cheekpieces of the bridle and the reins. THe mouthpiece is first pulled back in the mouth, then more pressure pulls it up. The curb chain keeps the action on the bars, and sends the pressure then up into the cheekpieces of the bridle, exerting pressure on the poll. This leveraging action can be quite harsh.
Steering with a shanked bit is not as subtle. The pelham, kimberwicke, and reining type bits have jointed shanks so that pressure can be exerted on one side of the mouth more easily, but western curbs with fixed shanks are suitable for only neckreining.
A full cheek bit, with keepers attaching the cheek piece of the bit to the cheek piece of the bridle, provide that same leverage effect. As the leveraging mechanism is much shorter, the pressure is more subtle, but can be quite effective.
A full bridle, with bit (the curb) and bradoon (the snaffle), provides all the action of each bit independently of each other. Modern riders tend to ride "on the bradoon," relying on the snaffle and using the curb only when necessary for extreme collection. The pelham, while double reined like a full bridle, does not provide this independent action and is vastly less subtle.
Other mouthpieces are variations on the theme... rollers, waterford joints, twisted wires, double twisted wires, and so on. I could write and encyclopedia, but I've probably bored you enough already!
Oh, well. Here's a great website, that even has animated diagrams of how the bits work in the mouth:
http://www.sustainabledressage.com/tack/bridle.php
I don't agree with everything they say, but the diagrams are excellent!