How do antibiotics work?

First up, antibiotics are effective against bacteria. They don't try to kill, say, viruses, which is one reason why microbiologists get so worked up about doctors prescribing antibiotics for colds. (Colds are almost always viral in origin, especially initially, and antibiotics have absolutely no effect.)

There are three main ways that antibiotics reap their destruction.

1. They may work by impairing the synthesis of the bacterium's cell wall. Penicillin, the first antibiotic to be discovered, works in this manner, and so do many other antibiotics. Basically, a cell wall acts to protect the bacteria from the effects of osmosis. Since there is a very high concentration of molecules inside the bacterium relative to outside it, water would flood into the cell and burst it. This tendency to rupture is resisted by the strong cell wall, so obviously impairing cell wall synthesis is an effective way of killing bacteria.
2. The next way in which antibiotics work is by impairing the bacterium's synthesis of proteins. Since proteins are so vital to life (they make structures, including receptors, as well as the enzymes the control just about every biological process), it isn't hard to see how antibiotics of this class could be successful. Examples include the tetracyclines and the aminoglycosides.
3. The last way to mess with a bacterium is to impair the synthesis of the bacterium's nucleic acids (DNA or RNA). Again, since nucleic acids are the instructions to do anything the cell is capable of, sabotaging this is sure to work in our favour in the war. This group includes metronidazole and the quinolones.

Within each class, there are numerous individual antibiotics, many of which work in slightly different ways. For instance, amongst the second group, tetracyclines interfere with tRNA function whilst the aminoglycosides interfere with mRNA function. Nonetheless, the above offers a nice overview, I think.

Also, not all antibiotics kill the bacteria. Those that do are called bacteriocidal, but there are others that simply inhibit the bacteria's ability to multiply and wait for our natural immunity to finish off the rest. This type of antibiotic is called bacteristatic.