To immunize someone is to give them a vaccine that protects them against disease. A child's pediatrician can immunize her against measles, polio, and mumps, among many other illnesses.
Vaccines work by arming your immune system against being infected — in other words, they produce immunity, or immunize you. In some cases, being exposed to a disease can also immunize you; if you've had chicken pox, you are most likely immune to it and won't get it again. There is also a legal meaning of immunize, "To make legally immune," or "to protect from being prosecuted in court." The Latin root, immunis, means "exempt" or "free."