Five of our big 6 plastics are known as addition polymers. Addition polymers have a single monomer (homomonomer) that continually adds to itself over and over. In a general sense it is like this (assume the monomer is "A"):
So the chain increases one unit at a time. What is not shown in the simplistic scheme above is the WAY in which the bonds are made. That is where a specific mechanism comes in. Below is the mechanism for a radical (R·) initiated polymerization of pretty much any ethylene derivative. For us that means LDPE, HDPE, PVC, PP, and PS. Yes, all 5 of those plastics/polymers follow the general mechanism below. Note that the symbol "X" is used in the key position that makes each of these monomers different. Those groups were explicitly shown in the previous section.
We need a radical to get the show started. This can happen in a myriad of ways but I'll keep it simple and we will just let and incoming photon (like a UV one strong enough to break a bond) split (homolytically) a bond to yield two identical free radicals.
Now the reactive free radical comes into the ethylene unit (opposite end of any substituted groups - the end with 2 H's. The nature of the free radical (it wants another electron and to form a bond) as it approaches the C it pulls one electron from the double bond (specifically the π-bond) to form a single covalent bond to that carbon. The remaining electron from the now broken π-bond is in an unpaired p-orbital, thus making the C on the end now a free radical. So a new free radical is born (one monomer unit longer than before) and step 2 can now repeat thousands of times.
Like making a radical, there are a few ways that termination can occur. The easiest to imagine (and it DOES happen) is that any two free radicals in the reaction mixture can find each other and make a happy single bond. This is the opposite of the first step but pretty much ALL the free radicals at this point are very long polymer chains with the free radical functionality on the end.