A hornet is a large insect which has a relatively large sting. When you refer to a problem as being a hornet’s nest, what you mean is that it is one that causes people to become angry. When you `stir up a hornet’s nest’, you are asking for trouble; you say or do something controversial which results in a lot of commotion.
*The teacher’s article in the local paper stirred up a hornet’s nest. *The Minister’s remarks about the striking students stirred up a hornet’s nest.