An antioxidant is not actually a substance; it is a behaviour. Any compound that can donate electrons and counteract free radicals has antioxidant properties. Free radicals are a natural by-product of the body turning food into energy. The body has mechanisms to deal with these free radicals, because they are highly dangerous if left to their own devices.
Excess free radicals are a problem because they attack the body itself, damaging key cellular molecules such as DNA. Cells with damaged DNA may be more prone to developing cancer. Free radical activity has also been implicated in many other degenerative conditions.