When the chain has "stretched" it will no longer sit properly on the teeth found on the sprockets and chainrings, symptoms of this point being reached include the chain jumping off the chainring when sufficient pressure is applied to the pedal. This usually happens when you're trying to pull away after stopping (eg at a junction). Not only is this a pain, but it can also be both dangerous and painful (my foot tended to come off the pedal and the calf/shin smacked into the teeth on the chainring when this happened to me).
The problem with leaving the chain until it gets to this state is that it also wears the sprockets at the back and the chainrings at the front. If you allow this to happen, then even buying a new chain won't do you much good as the worn teeth on the other parts of the drive system will prevent the new chain from sitting properly, and may cause the same kinds of problems as were being experienced with the old chain. If you're lucky enough not to have a problem with the chain skipping, the worn teeth will still cause the chain to wear out much faster than normal.
So, check you chain fairly regularly, if as mgb says, the chain stretched by more than 1/8 of an inch per foot, change it. Otherwise you might have to replace either the chainrings, the rear cassettes or both. A much more expensive proposition all round especially if you've got a decent chainset fitted.