It is always hard to come up with a definition of these things without either being too sciency and hard to understand, or dumbing it down too much and sounding like you are talking to children. The latest health article in The Guardian I think explains it perfectly, so I will say no more…
“Free radicals are unstable elements that come spinning off any oxygen-using chemical reaction in the body. They are unstable because they are missing an electron and, in a bid to restabilise themselves, they steal an electron from elsewhere. This could be from the fats in cell membranes or from your DNA. The damage they do when bullying other elements into handing over an electron is called oxidative stress, and this can be associated with heart disease, cancer and diabetes.
Free radicals, however, are also involved in beneficial processes. They help to destroy invading bacteria and play a part in cell communication. To limit their role to only those things that benefit us, our bodies make things called antioxidants that, much like people standing outside nightclubs handing out hugs and hot chocolate to pacify drunken revellers, provide free radicals with the electrons they need so they don’t cause damage elsewhere.”
One thing I will say is that you should have a read of the article “the truth about ‘miracle’ foods – from chia seeds to coconut oil“. It talks about not removing antioxidants from their context of the whole food – “the nutrients in many foods become available to us only when eaten as part of a wider diet”. This is an important thing to remember when you find yourself staring at shelves packed full of individual antioxidants and vitamins. Unless there is a particular deficiency or need for large quantities of one nutrients, eating the whole vegetable/nut/fruit/seed instead will give you fibre, protein, energy and other nutrients that you don’t get in the bottle.