A polymer is a substance made up of a repeating sequence of one or more types of units, or monomers, which are bonded together to form a chainlike structure.

Polymers made up of one type of monomer unit are called homopolymers e.g. A-A-A-... and polymers made up of more than one type of monomer unit are called co-polymers e.g. A-B-A-B-....
Polymers can have different properties depending upon the type of monomer unit, the number of monomers in the polymer, how the monomers fit together and whether the monomers have any additional chemical groups. They can be elastic, durable, flexible, hard, soft, solid or liquid.
Polymers are essential for our survival; for example, DNA, starch and protein are all found in the human body.
Polymers can be found in nature, such as mushrooms and shellfish.
There is an enormous variety of man-made polymers; from clingfilm, to dental resins to fix teeth, to novel lightweight aeroplane parts which make aeroplanes more lightweight and more energy efficient.
A plastic is a type of polymer. Plastics are defined as synthetic, water-insoluble polymers that are repeatedly moulded, extruded or physically manipulated into various solid forms which retain their defined shapes in their intended applications during their use and disposal.
However, as can be seen from the examples above, a plastic is a type of polymer, but not all polymers are plastic.
Essentially, plastics are man-made materials which are made from a wide range of organic polymers that can be moulded into a specific shape while soft, and then set into a rigid or slightly elastic form. The properties of a plastic can be affected by the number of single units in the polymer structure (known as monomers) and how they fit together.