What are vaccines?
There are many diseases that, if you catch them once, you will never catch again. Measles is a good example, as is chicken pox. What happens with these diseases is that they make it into your body and start reproducing. The immune system gears up to eliminate them. In your body you already have B cells that can recognize the virus and produce antibodies for it. However, there are only a few of these cells for each antibody. Once a particular disease is recognized by these few specific B cells, the B cells turn into plasma cells, clone and start pumping out antibodies. This process takes time, but the disease runs its course and is eventually eliminated. However, while it is being eliminated, other B cells for the disease clone themselves but do not generate antibodies. This second set of B cells remains in your body for years, so if the disease reappears your body is able to eliminate it immediately before it can do anything to you.
A vaccine is a weakened form of a disease. It is either a killed form of the disease, or it is a similar but less virulent strain. Once inside your body your immune system mounts the same defense, but because the disease is different or weaker you get few or no symptoms of the disease. Now, when the real disease invades your body, your body is able to eliminate it immediately.
Vaccines exist for all sorts of diseases, both viral and bacterial: measles, mumps, whooping cough, tuberculosis, smallpox, polio, typhoid, etc.
Many diseases cannot be cured by vaccines, however. The common cold and Influenza are two good examples. These diseases either mutate so quickly or have so many different strains in the wild that it is impossible to inject all of them into your body. Each time you get the flu, for example, you are getting a different strain of the same disease.
What are antibiotics?
Sometimes your immune system is not able to activate itself quickly enough to outpace the reproductive rate of a certain bacteria, or the bacteria is producing a toxin so quickly that it will cause permanent damage before the immune system can eliminate the bacteria. In these cases it would be nice to help the immune system by killing the offending bacteria directly.
Antibiotics work on bacterial infections. Antibiotics are chemicals that kill the bacteria cells but do not affect the cells that make up your body. For example, many antibiotics interrupt the machinery inside bacterial cells that builds the cell wall. Human cells do not contain this machinery, so they are unaffected. Different antibiotics work on different parts of bacterial machinery, so each one is more or less effective on specific types of bacteria. You can see that, because a virus is not alive, antibiotics have no effect on a virus.
One problem with antibiotics is that they lose effectiveness over time. If you take an antibiotic it will normally kill all of the bacteria it targets over the course of a week or 10 days. You will feel better very quickly (in just a day or two) because the antibiotic kills the majority of the targeted bacteria very quickly. However, on occasion one of the bacterial offspring will contain a mutation that is able to survive the specific antibiotic. This bacteria will then reproduce and the whole disease mutates. Eventually the new strain is infecting everyone and the old antibiotic has no effect on it. This process has become more and more of a problem over time and has become a significant concern in the medical community.