You've probably stood across the counter, or sat across the table from a customer that was having a bad day. It happens. If you're in business long enough someone will probably get upset about something. I mean, that's just how relationships work, and relationships between a business and its customers are no different.
Usually when the person we're talking to is upset, we react in one of two ways. We either want to do something to address whatever is making them upset, or we get defensive.
Maybe it was your fault. Maybe it wasn't. Either way, in that moment, it's your problem. What you do next is everything, and there's one thing you should never do-- or at least say.
An apology is almost always a good move, but let's be clear-- that's not an apology. You can't be sorry for the way someone else feels.
You should express understanding of their feelings, and you should validate them-- but apologizing for them does neither. It's condescending and puts people on the defensive.
Instead, say "I'm sorry I (or we) dropped the ball. I can see why that is so frustrating for you. Let me see how we can fix it."
Problems are often an opportunity to win a customer or client for life. How you handle when things go wrong is a chance to remind them that they are "the most important person in the world," at that moment if you treat them accordingly.
For some reason, your customer isn't happy. It's not because your tall or short, or have brown hair, or wear glasses. Things can get personal very quickly if you let them-- so don't. Stay professional. It makes it much easier to figure out what's really going on.
When you deal with the issue instead of allowing it to be personal, you have a chance to actually correct whatever went wrong. Before you act or respond, make sure you really understand what's going on. Ask questions like "I understand you're upset. Can you help me better understand what happened, and how I can help make this right for you?"