There are a few problems that can happen when your queen gets old.  She may slow down her egg laying.  She may start to miss cells when she is laying eggs (Spotty brood: topic 7). She also may start to lay more drone eggs than your hive needs.  As she gets older her pheromones (queen smell) gets weaker.  Sometimes this causes your bees to swarm.

If your queen is not working well, then your hive will not do well.  Sometimes your bees will see that their queen is failing and will begin to raise a new one.   Other times they will do nothing and the hive will do poorly with their poor queen unless you re-queen the hive.  That is why it is a good idea to replace your queen every year, before you have a problem.