Free radicals are chemicals with an unpaired electron. That electrons are negatively charged, and thus repel each other, yet tend to pair up in twos is a peculiar aspect of nature. You can read more about that paradox, in Why Do Electrons Pair if They Repel Each Other?
But first, what is an electron? Electrons are one of three ingredients that compose atoms.
Electrons tend to appear in twos on a molecule. They either comprise a bond or exist as “nonbonding” electron pairs. Atoms are the building blocks of molecules. Sometimes two electrons are shared between two atoms, being attracted to the positive protons of both atoms at once, and this sharing creates the bonds between atoms that hold a molecule intact. Even though the like-charged positive protons of both atoms’ nuclei repel, they are tied together by the two electrons like two dogs fighting over a bone. Every bond is made of two electrons, so we see that electrons form pairs in bonds.
Other electrons are not involved in this sharing and are called nonbonding electrons, which simply hang out on particular atoms in the molecule. These, too, occur in pairs, so are sometimes called nonbonding pairs or lone pairs. So, although electrons really repel each other, we still find them engaged in this remarkable pairing behavior. When a single electron occurs, it cannot pair, and this creates a free radical.
A free radical can be an atom, a molecule, or an ion, but along with its normal paired electrons, it must also have a single, unpaired electron. A free radical could be either an individual atom, the building block of molecules, or it could be a molecule, which is simply a bunch of atoms held together by bonds. It could also be an ion, which is either an atom or molecule with excess or missing electrons; so, unlike atoms, ions always have a charge of some sort. Atoms, molecules, or ions can have unpaired electrons, and when they do they are called free radicals, or just radicals. The adjective “free” is unnecessary but has become habitual, perhaps because it adds drama. The unpaired electron is sometimes represented in chemical drawings as a single, isolated dot on a chemical. For example, molecular oxygen, which is two oxygen atoms bonded together (O2), can gain an extra electron to become the noxious free radical called superoxide (O2-). (The added negative sign indicates it gained one unit of negative charge.)