The word comes from Greek: poly means `many' and mer comes from merous which roughly means `parts'. Our word polymeric comes directly from the Greek polumeres which simply means `having many parts'.
Polymers are large molecules consisting of repeated chemical units (`mers') joined together, usually in a line, like beads on a string. Each `mer' is typically made up of more than 5 and less than 500 atoms; the word `polymer' is applied when you have more than about 50 `mers' stuck together.
Some ubiquitous `mers' are ethylene, styrene and acrylamide. Each of these may be polymerized to make, respectively, polyethylene (the soft clear plastic that plastic bags are made of), polystyrene (the stiffer, usually white plastic that the covers for soft-drink cups are made of), and polyacrylamide (the very tough, clear plastic that compact discs are made from).
Look on the bottom of a recyclable plastic bottle - chances are you will see a PE or PS which means polyethylene or polystyrene. These materials are examples of what happens to polymers when they solidify: the chains are entangled and packed together to make light, tough, flexible materials.
A way to think about some of these materials is to think of what a big glob of cooked spaghetti is like. If you stretch it a bit, it is kind of elastic, but if you really pull hard, the noodles start to slide past one another and the whole glob starts to permanently deform. At least that is the idea! Does this remind you of what happens to a PE plastic bag when you stretch it? Think about what must be happening to the microscopic spaghetti that the bag is made up of!
If you heat up PE or PS to moderate temperatures, if the chains have not been chemically stuck together (`cross-linked') they will melt, and turn into goopy liquids, which are called polymer melts. Some polymers are melts even at room temperature, like polydimethylsiloxane (PDMS), or poly(ethylene-propylene) (PEP).
Remembering that paper is made of cellulose, which is a polymer of biological origin, if you look around the room that you are in, you will see that a good fraction of the stuff in it is made of polymers. And of course, you are, too!
Historically, polymers have mostly been used to make solid plastics where the chains virtually don't move. But nowadays people dream of new applications of polymer liquids where fluctuations (Brownian motion) and interactions (the sticking together or association of different types of molecules) can play a more important role. Many of the most important research problems involve polymers free to fluctuate about in a small-molecule solvent. Naturally, the most important solvent is also the hardest one to understand: water. An important area of research is the modification of the properties of surfaces using thin polymer coatings.
The sky is the limit for these wet technologies: living organisms are mainly composed of polymerized amino acids (proteins) nucleic acids (RNA and DNA), and other biopolymers. The most powerful computers - our brains - are mostly just polymer glorp soaking in salty water!
Suppose you have a bunch of micron-size spheres that you want to suspend in a solvent (e.g. water) to make a colloid - maybe to make a paint (little latex spheres), a medicine (small biodegradable plastic spheres impregnated with a drug), a magnetic fluid (small magnetic particles). You will certainly have a problem - the little spheres will glom together thanks to van der Waals attractions, and you will have a glob of yecch at the bottom of your bottle, instead of a nice stable colloid.
What you need is some kind of repulsive force field acting between the particles to push them apart. A way to do this is to end-attach a bunch of flexible polymers to the surface of the spheres to make a molecular forest - or a polymer brush. The polymers stretch out away from the surface and away from each other in order to make intimate contact with the solvent, pushing the particles apart, and stabilizing the colloid. This technology is used in the real world.