Formation of polymers - addition polymerisation
During radical polymerization, we use molecules called initiators $I$ for forming free radicals. The molecules of initiators contain  at least one fragile bond that can easily dissociate upon heating, and yield free radicals.
$$I\rightarrow2A^.$$
The as-formed radicals  have enough energy to attack the monomer molecule $M$ and transfer the radical to the monomer molecule:$$A^.+ M\rightarrow  AM^.$$
More precisely, one electron of the double bond in the monomer forms a covalent bond with the free electron of the radical $A^.$. The other electron of the double bond is now free. We have  a radical  on the monomer  $AM^.$. 

The radical monomer  attacks another monomer molecule, and the polymerization reaction propagates:
$$  AM^. + M \rightarrow AMM^.  $$
In the above reaction, one electron of the double bond in the monomer molecule forms a covalent bond with the free electron of the radical $AM^.$.The other electron of the double bond is now free $AMM^.  $. 
and so on..