The free-radical theory of aging is that organisms age because cells accumulate free radical damage with the passage of time.
In general, a "free radical" is any molecule that has a single unpaired electron in an outer shell. While a few free radicals such as melanin are stable over eons, most biologically-relevant free radicals are fairly reactive. For most biological structures free radical damage is closely associated with oxidation damage.
Oxidation and reduction are redox chemical reactions. Most people can equate oxidative damage with the process of rust formation of iron exposed to oxygen. Oxidation does not necessarily involve oxygen, after which it was named, but is most easily described as the loss of electrons from the atoms and molecules forming such biological structures. The inverse reaction, reduction, occurs when a molecule gains electrons. As the name suggests, antioxidants like vitamin C prevent oxidation and are often electron donators.