Electrons almost always exist as pairs in stable chemical bonds. A radical holds one single electron that desperately needs a partner. This lonely electron makes the radical steal an electron from a nearby molecule.
That theft creates a brand new radical and starts a rapid chain reaction. Heat or ultraviolet light often provides the exact energy needed to break bonds evenly. Chemists call this perfectly even split a homolytic cleavage.
Radicals play massive roles in many natural and industrial chemical processes. They help connect small molecules into the long plastic chains found in everyday items. They also drive the rapid burning of gasoline fuels in car engines.
High in the atmosphere, radical reactions constantly create and destroy the protective ozone layer. Inside the human body, biological oxygen radicals can cause severe damage to healthy cells. This cellular damage slowly contributes to the aging process and many common diseases.
You might hear people talk about eating antioxidants to stop this exact damage. Antioxidants safely give up an electron to neutralize the radical without becoming dangerous themselves.