Polymerization is the process by which monomers react together chemically in order to form a polymer chain. There are many different forms of polymerization, a few of which will be outlined here.
In addition polymerization, a catalyst initiates the polymerization in a solution of monomers. The catalyst turns a small fraction of the monomers into a chemically reactive form, by breaking a double or triple bond contained in the monomer. These chemically reactive monomers combine with other monomers (thereby making them chemically reactive) to form a chain. Each time this happens, the chain ends remain chemically reactive. The reaction is stopped by adding a chemical that inactivates the catalyst's effect on the chain ends.
In condensation polymerization, many chain ends may react with one another. Two molecules join together, which results in the loss of a smaller molecule (often, this smaller molecule is water). The end chain structure is dependent on the number of functional end groups of the monomer which can react.
In living polymerization, the polymerization event is reversible, thus making it possible for polymer chain to exchange monomers freely amongst themselves. It is a special form of addition polymerization, where the ability to terminate a growing polymer has been removed.
 T. Witten, "Structured fluids," Oxford University Press Inc. (2004).