Long, tangled polymer molecules that have a few cross-links between them can extend and then regain their original shape. These stretchy materials are called elastomers . The only problem for the serious balloon collector is that the open mesh structure of latex, particularly when it's stretched tightly, lets helium and other gases escape right through the tangled molecules. Mylar, a type of polyester that can be rolled into thin films, is better at trapping helium because its molecules are more closely packed and Mylar does not stretch out like latex. But even a Mylar balloon can't hold all the tiny helium atoms forever, and eventually it loses lift also.
Balloons are a great example of how the pressure and the volume of a gas are interconnected.
When you blow up a balloon, you exert pressure on the inside walls of the balloon. When that pressure exceeds the outside air pressure plus the pressure exerted by the latex itself, the balloon begins to expand. The pressure inside a balloon is always a little higher than the surrounding air pressure, because the latex is pushing back as the air inside pushes out. When a weather balloon rises in the atmosphere, for example, the outside pressure decreases and the balloon expands. Eventually, the inside pressure causes the balloon to burst.