There will be many situations where people are unpleaseable, so don't allow difficult customers to bring you down. There will be days when you will have to deal with them so be prepared. First, listen to what they have to say. They may have a legitimate complaint to make and want some answers. Once they see you are willing to assist with the situation they will change their tune.
Sometimes though you just can't do enough to make someone happy. They can threaten to run your business name through the mud and even to talk to you terribly. Remain calm and le them know you will be glad to help them with their problem once they have had a chance to calm down. Ask them to call you back or if they would like you to return a call.
You may have to become firm in order to get a customer to see you won't be pushed around though. There is a difference between being flexible and helpful and them just backing you into a corner. You don't want to compromise the integrity of your business because of their issues.
Make sure you and your staff don't take such outbursts from difficult client's personal though. If you feel you have done everything possible to remedy the situation then you should feel good about it. Difficult people are a part of life. It's how you deal with them that will keep you on top of the game and make you the better person.