There are tons of different types of polymers. Their properties vary not only based on their molecular structure, but also based on many other superstructural and external factors. The poly-dispersity (the range of polymer sizes, usually by MW), the addition of non-polymerized external molecules (so-called plasticizers) which are used to tune the polymer for hardness, flexibility, etc., and the overall denisty of the polymer are also important. You can also have copolymers where you have polymerized two different monomer units into one polymer. These can be either alternating monomer units, random, or block copolymers where the two (or more) units are polymerized in separate "blocks" along the larger polymer. You can also make cyclic polymers where you make one huge polymer ring; these often have different properties than linear polymers of similar size. Cross-linked polymers are very important. I cross-linked polymers, linear polymer chains are attached to one another through chemical bonds. These are often very rigid polymers.
On a molecular level, the stereochemistry of nearby atoms can have large effects on the macroscopic behavior of the polymer. Generally you can have either random stereochemical arrays, alternating R/S, or single stereochemistry (e.g. R only) polymers. Some of the macroscopic properties of polymers can be predicted based on known polymers with similar structures, but again there is a lot of trial and error involved in finding the kind of polymer you want!