The type of bridge you use determines how the strings' vibrations transfer into the body. Usually, choice of bridge has more to do with comfort and musical style than it does with tone, but there are some characteristics worth noting. First, the bridge needs to be made of very hard wood, bone or metal in order to transfer the strings' energy rather than absorb it. You don't want your bridge filtering those vibrations, that's the wood's job. Second, it needs to be securely fastened to the guitar's body with bolts, screws or glue.

The Gibson Tune-o-matic bridge is the classic combination bridge and tailpiece. Both parts have height adjustment bolts that are screwed into nuts which are sunken into the body. The actual bridge and tailpiece are held onto these bolts by the tension of the strings, which is ample to create a solid connection.

Bridges that I call "flat" have a low profile and flat bottom. They are usually secured with four to six screws and have a large contact area with the body. You would think this makes it more secure than the bridge and tailpiece, but its saddles are also held down merely by the tension of the strings.

A bridge that is able to tilt, thereby "bending" the strings is a tremolo. Most of these are held on two special bolts by the tension of the strings on top, as well as the counter-acting tension of springs in a cavity behind the guitar. Trems are known to have a thinner sound than more conventional bridges. They are built to move, so they do absorb some of the strings' energy. Evidence of this is the fact that when you play certain notes on a guitar with a trem and then mute the strings, you can hear the springs vibrating sympathetically. Pickups can hear that. If you desire a thick sound as well as the finesse of a trem you can cope by choosing bass-heavy pickups and wood.

A long time ago I was in a local guitar shop and was told by a salesman that putting the strings through the body (instead of a tailpiece) makes the guitar sound "way better". This is BUNK. Strings should not vibrate past the bridge or the nut. If they do you will notice intonation problems. So the strings aren't actively transferring energy through the tailpiece or ferrules (the little metal pieces installed in the holes that hold strings in the body).

In fact, strings going through the body do the exact same thing as strings through a tailpiece: change the tension points on the body. Strings pull up on the tailpiece and push down on the bridge. A bridge without a tailpiece (or strings through the body) does not have that upward tension behind the bridge. How does that change the tone? Good question, as in, "I haven't the foggiest." If you recall the sand-on-a-vibrating-plate idea from Guitar Tone Part 3 - The Body, these various tension points will affect the vibrations of the body and possibly create new nodes in the body's vibrations. This changes the tone, but there will likely never be a scientific study done that uses many guitars that are otherwise identical except for having different bridge types and tension points. I'll just keep guessing until then ;-)