We all run into this from time to time. It can be as a customer speaking to an employee. It can be as an employee speaking to a customer. Whatever your position, what you do about it, how you handle it is what's most important.
Diplomacy is the verbal art of pleasing most of the people most of the time.
It's very important all of the time, but especially when it comes to Customer Service. Diplomacy is not about being insincere. It's not about saying something just because you think it's what the other person wants to hear. When you're able to be calm under the somewhat stressful situation of dealing with an unhappy customer, the customer senses you are confident and self-assured. Customers need to get this sense of confidence and self-assurance from you.