A polymer, as defined by the Funk and Wagnalls Standard Dictionary is:

Polymer - n, A compound of high molecular weight formed by the chemical combination of two or more molecules of the same kind.

Which is true enough, as far as it goes, but the definition is just general enough to be largely useless. So lets consider an example: take an ethylene molecule (two carbon atoms joined together, with two hydrogens on each). If I do that chemical voodoo that they (chemists) do and join a whole bunch of these together, I get:

—[—CH2CH2—]— Poly(Ethylene)

Of which you may have heard (the little brackets around the ethylene are polymer person shorthand for "repeat this a whole bunch of times"). Or, suppose I pop one of those hydrogens off and substitute a chlorine atom. Then I get

—[—CH2CH —]— Poly(vinyl Chloride)
                  |                                     PVC

Of which you have almost CERTAINLY heard (it's the PVC in PVC pipe, which have almost completely replaced metal pipes when water is to be moved from one place to another). Or maybe you can...well, you get the idea ("I've got one word for you, Alex. Plastics."). By putting in different compounds and combining them in clever and skillful ways, you can make egg cartons and milk jugs and bullet proof vests and Porsche fenders. But that's not even the whole story. We have considered only polymers built around carbon. And there was nothing in Funk and Wagnalls that limited us to carbon. Suppose we take amino acids and string them together one after the other. Then we have:

—[—amino acid—]— polypeptides, proteins, DNA, RNA

In other words, the molecules responsible for maintaining and propogating life are polymers. The examples we showed from the plastics industry were repeats of only one molecule (whether ethylene or vinyl chloride).  This is called a homopolymer.  If we repeat two or more different molecules (like amino acids) we call that a copolymer.  And so all the things we can do to try and understand the plastics industry are fair game to try and understand aspects of biological polymers. In addition, complicated living things (like oak trees and humans) store carbohydrates in long chains (maybe to store sugars for a rainy day - like starch in a potato - or maybe to use the sugar to build a tree trunk - like cellulose). Then you have:

—[—carbohydrate—]— polysaccharides, starches, cellulose

A particular example of this is the molecule xanthan, which is a particularly stiff polysaccharide used as a thickener (look at a the ingredient list on a bottle of salad dressing sometime). I study xanthan to understand why it is so stiff and how the molecules interact. And even though poly(vinyl chloride) and xanthan are both polymers, one obviously makes a better thickener in salad dressing!