Polymerization is the process of building big molecules out of small ones. You're doing it right now as your body builds proteins out of the food, gases and liquids you take in. There are numerous ways that such reactions can be started and they generally involve the application of energy. When light is the energy source, the process is called photopolymerization.

Composition
Photopolymers are basically a soup of ingredients that work in concert to make this process happen. They may contain a number of components, but there are just three main ones:

1) Binders or Oligomers

This material consists of long, chain-like, chemically-reactive molecules that give the final solid its mechanical and other properties. Examples include acrylates, epoxies and urethanes. Most of a photopolymer consists of these binders which might typically be in the range of 50 to 80% of the total weight.

2) Monomers

Monomers are small molecules that also enter into the reaction. They're used mainly to lower the viscosity because a soup made from binders alone would be very thick and not easy to spread or otherwise handle. Some monomers are multifunctional offering more than a single chemical route to polymerization. Some examples include vinyls and shorter acrylate molecules. Monomers may typically constitute 10 to 40% of the photopolymer.

3) Photoinitiators

Photoinitiators are molecules that can be split into two or more parts by exposure to light. At least one of these parts is capable of reacting with both the monomers and binders to link them together. Photoinitiators are only sensitive to specific wavelengths of light. While there are initiators that work with visible light, most of the ones used in additive fabrication are sensitive to ultraviolet radiation. The spectrum of the light source used must overlap that for which the initiator is sensitive. Both lasers and other types of light sources such as arc lamps may be used. Photoinitiators comprise just a few percent of the photopolymer mixture.

Changing Liquids into Solids
There are two major chemical schemes that are commonly used in photopolymerization, each requiring a specialized photoinitiator. In free radical polymerization a complex ion (radical) is split off the initiator by the light radiation and that combines with a monomer or binder molecule to start the reaction. In cationic polymerization, light causes a strong acid to be released by the initiator and that in turn starts the bonding process. Most photopolymers today are based on the free radical scheme, but more advanced materials use cationic polymerization which frequently offers better final solid material properties.

During the reaction, the molecules of binder and monomer not only combine to make longer chain molecules, but they also bond from chain to chain in a cross-linking process. Both the composition of the binders and monomers used and the way in which they bond together are responsible for the properties of the final solid material.

Once the polymerization process starts, it keeps right on going because it's a chain reaction. Each time monomer or binder molecules link to form a gelling solid, they liberate other radicals or acid molecules that in turn cause additional molecules to bond - and so on. In some cases the propagation can even proceed in the dark for a short time after the light source is removed.

The chain reaction continues to propagate until radicals recombine to form non-reactive products, or the ingredients needed for the reaction are no longer available in the correct proportions, or finally by trapping reactive molecules in the hardening matrix where they can no longer move into position to combine.

Using a chain reaction in the polymerization process is highly advantageous. It means that a relatively small quantity of light radiation energy can cause a large amount of the liquid photopolymer to solidify. The net result is that the effect of the light is multiplied hundreds of times. Nevertheless, not all the material is converted. Some free monomer is always trapped in the solid matrix. This can cause the properties of the solid to change over time and is an area where improvement continues to be made. In many cases "baking" or flooding the final part or object in a box where it's subject to intense UV light will effect a more complete cure and more stable properties. This is often referred to as post-curing.

The process of changing from a liquid to a solid provides many opportunities for errors and other problems. During the transition photopolymers may undergo shrinkage resulting in dimensional inaccuracies and part distortions such as curl. However, great progress has been made both in materials and the processes themselves over the years so that now these effects have been minimized or controlled. Today most photopolymer research is aimed at widening their properties to mimic engineering plastics more closely, or to provide innovative functionality.

This description is but the tiniest tip of a very large iceberg. You can learn much more from the references, and other deep resources such as the patent literature.

References:

Jean-Pierre Fouassier, J. F. Rabek; Lasers in polymer science technology: applications; CRC Press, 1989; pp 2-24.

Joseph C. Salamone; Polymeric Materials Encyclopedia; CRC Press, 1996; pp 5181-4

Patrick Glöckner et al; Radiation Curing; Vincentz Network GmbH & Co KG, 2009; Sec. 2.1 to 3.3.

V. V. Krongauz, A. D. Trifunac; Processes in Photoreactive Polymers; Springer, 1995; pp 3-33.

Paul F. Jacobs; Rapid Prototyping and Manufacturing: Fundamentals of Stereolithography;
       Society of Manufacturing Engineers, 1992; Chapter 2.