While many oxidation reactions occur naturally in humans, our bodies are constantly trying to maintain a reductive environment. Failure to do so is known as oxidative stress. When one is in oxidative stress, oxidizing agents such as peroxide form in the body through reduction reactions. These oxidizing agents can produce free radicals, which are single unpaired electrons. Free radicals cause chain reactions, whereby the product of the reaction is very reactive itself (due to the passing of the free radical). These chain reactions are not desirable and can cause cell damage. Antioxidants do two things. First, they are reducing agents and can be oxidized themselves without causing free radicals. Second, they can also remove free radicals from the body and thus terminate the chain reactions.
Antioxidants are therefore a good tool for avoiding oxidative stress. Antioxidants are used in many methods of disease prevention and treatment, and are considered to be a critical part of a healthy diet. However, overconsumption of antioxidants can also be bad. Antioxidants can be found in a variety of foods, but primarily in fruits and vegetables.
More information on the health effects of antioxidants can be found here: http://www.nlm.nih.gov/medlineplus/antioxidants.html