Free radicals are molecules that possess an unpaired electron in them.
They have the tendency to undergo reactions to form a paired lone pair on the molecule. Due to this, it takes in the electron from another molecule by breaking the bonds, forming a chain-like process.
Free radical chain reactions are those reactions which involve the formation of free radicals as intermediate during the reaction by homolytic cleavage of bonds.
It is a three-step process which includes initiation, propagation and termination. In initiation, involves the formation of the free radical species, by breaking the bonds or exposing molecules to UV rays etc.
The chain part of the process is in the propagation process, where the earlier formed radicals react with new molecules to form new radicals.
The termination has the free radicals reacting with each other to form a non-radical molecule.
An example of this is the halogenation of alkanes.