"The answers seem simple - time, temperature, pressure, co-monomers and catalysts/other active chemicals - but their application can be quite complicated. Regarding length, the longer the mass is at reaction conditions, the more of it will react, but there is a diminishing-returns principle acting when the mass is, say, 95% polymerized, and the remaining 5% of monomer is looking for a loose end to hook up with. Some polymerization, notably PVC, can be done in water suspension, which enables more free movement of monomer and short chains looking for others, but requires a draining and drying step at the end of the line.While the question is only marginally answered (the answer doesn't really describe how to control molecular weight, and controlling branching is given in only in the context of olefin copolymerization), even within the answer itself, there are sections that I take issue with, and these were highlighted in bold.
The role of co-monomers is shown with linear-low-density PE. To get short branches, a small amount of another monomer with a double bond such as butene (4 carbons instead of the two of ethylene) is included. The butene goes into the chain with its 2-carbon double bond just like ethylene, but that leaves 2 carbons in a short chain (the branch) dangling from the main chain. Similarly, hexene (6 carbons) makes 4-carbon branches and octene (8 carbons) makes 6-carbon branches.
There are many more complications but this addresses the basics of your question."
First, to more properly answer the question: Molecular weight can be controlled by the adjusting the amount of initiator (more initiator leads to more chains with smaller molecular weight), addition of chain-transfer reagents (more of it will stop the chains short), and adjustment of reaction conditions such as time, temperature. Branching can controlled by the addition of polyfunctional monomers, appropriate catalysts, reactor design or as Allan suggested, by use of select comonomers. And also as Allan noted, "There are many more complications..."
But turning now to the answer Allan provided and the sections I bolded, the idea that the remaining monomer needs to find the growing end of the chain is correct, but this model is oriented pretty much exclusively towards free-radical (addition) reactions, and not condensation reactions. In the latter reaction such as between a diacid and a diamine to form a polyamide, the monomers disappears very quickly - one of the comonomer merely needs to find one of the other comonomer to react with and boom, they're gone. The situation of having 95% polymerized material and 5% monomer speaks to a free-radical reaction where only a certain number of monomers were initiated and the chains grow only from them. Certainly a larger volume of commercial polymers are prepared by a free-radical reaction (all the olefins and PVC for starters), but without condensation reactions, there would be no nylons, polyesters, polyurethanes and others - a group of polymers too large to ignore, even in this small space for answering.
Regarding water suspension polymerization, the water does not "enable more free movement of monomer", at least to the extent of aiding the kinetics of polymerization. The reaction kinetics of suspension polymerization are well documented as being equivalent to bulk polymerization and thus limited in the manner previously noted. The water does allow for freer movement within the reactor, but once the monomer has moved within the suspension particle, the water is absent and the diffusion of the monomer to the reactive end is again the limiting issue.
I'm not sure that clarifying any of this will help an engineer with extruding a plastic, but it certainly doesn't hurt to know (and and to know it correctly).