Cistern keeps running and doesn't refill?

Our low-level, ordinary siphon-type cistern has an intermittent fault which has been happening for the last few weeks. The toilet flushes normally, but sometimes after it finishes water continues to flow into the toilet bowl, and when you look in the cistern, the water level is very low. It is refilling, but emptying into the toilet again at about the same rate. Pressing the flush lever again usually sorts it, and my daughter noticed that if reflushing produces a gurgling sound then the cistern will refill properly but if no gurgling noise it will continue with the same problem. I will be grateful for any help in diagnosing the problem, and suggestions for fixing it. Many thanks in advance.