We can't exist without free radicals as they are important for certain reactions within the body, for example the immune system creates them to neutralise viruses and bacteria. However when too many are produced through activities such as eating fatty food, smoking and drinking it causes an imbalance - it's the imbalance that is thought to contribute to diseases such as cancer, stoke and diabetes.
Free radicals are organic molecules which have an uneven number of electrons making them unstable and highly reactive, always on the look out for an electron to steal from its nearest neighbour. They are called 'free' because they float around until they find a way to stabilize. When the 'attacked' molecule loses it's electron, it then becomes a free radical itself beginning a chain reaction. In the body this results in cellular damage as the electrons are stolen from the cells of healthy tissues.
One way to feed the hungry electron-appetite of free radicals is to eat more antioxidants - antioxidants neutralize free radicals by donating one of their own electrons - they don't become free radicals themselves because they are stable in either form thereby halting the electron stealing cascade. Ensure that you get an adequate intake of antioxidants by eating a balanced diet consisting of plenty of fruit and veg or consider taking a supplement.