Rubber bands are made of polymers. Polymers a really big molecules (often called macromolecules) which are made up of many many small molecule repeat units known as monomers.

The reason rubber bands stretch is because when you pull on the rubber band, the polymers (which are normally coiled up more or less at random) get stretched lengthwise and the rubber band is lengthened. When you stop pulling on the rubber band, the polymers return back to their more coiled state and the rubber band goes back to being shorter.

One cool experiment which you can easily do with a rubber band is to take a rubber band and hang a small weight off of it. This will cause the polymers to stretch and the rubber band to lengthen. Next, if you take a hair dryer and heat the rubber band, the heat will cause the polymers to want to coil back up and you can watch the rubber band then shrink and pull the weight back up a little bit.

Rubber bands are made of natural rubber. Natural rubber is made of long chains of molecules called polymers. Because these polymers are so long (they can be thousands of molecules long) they get tangled up in themselves. The result is a property called elasticity, the polymers are elastic. This is why rubber bands are sometimes called elastic bands. When you stretch the rubber, the polymers start to elongate or stretch. This requires some force. When you let go, the polymers relax back into their tangled compact shape. Most plastic materials have some elasticity, but the amount the material will stretch depends on the type of polymer and length, etc.

The ability to stretch is what we call elasticity (elasticity is the noun, elastic is the verb). Rubber is elastic because the chemical bonds that hold it together can store energy in the form of tension and allow the rubber to stretch, Beyond that I don't think I'm qualified to explain.