A timing chain does the same job as the timing belt (cambelt). It drives the camshaft(s) in an engine, the timing of this process is vital and must be within a very narrow range to allow the engine to run correctly. Timing chains need replacing at certain intervals (depending on the car) and can wear out quicker if engine oil levels aren't maintained.
Timing chains can stretch with wear, they can also stretch due to sudden shocks such as mishifting a gear. When the chain is stretched the timing of the engine will be out and will lead to all kinds of problems. Badly stretched timing chains can lead to catastrophic engine damage.