How we change what others think, feel, believe and do
If a mistake occurs, and in a busy restaurant this is quite possible now and again, the bottom line is to get it corrected as quickly as possible.
Typical restaurant mistakes include serving the wrong order, serving food that is not hot enough and dropping food (sometimes on the customer!).
Apologies and immediate correction of the problem is often enough for most people. When people are upset, bringing the manager over can help also. If they continue to be upset, ask them what they would like you to do.
Tipping food on them can be a problem. An apology and 'Are you ok' for small splashes of food may sufficient, but a larger dump may mean you must immediately offer to have the clothes cleaned.
A standard way of apologizing is to apply a discount to their bill that is proportionate to the mistake or giving them something like a free drink. Another way to help them return again is to give a 'free dinner' voucher.
Is apologizing admission of liability? Well, that may be something for the lawyers, but what is often true is that an apology defuses the situation and will calm most people. By and large, customers are out for a good time and do not really want an argument.
Arguing the point, even if you are right, is seldom a good idea as you will likely either upset the customer further or they will get angry, upsetting everyone in the restaurant.
If you are a waiter, then a bad mistake (or even a quite simple one) can get you sacked, though only if the management know. Managers are likely only to know if the customer tells them, so immediate action on your part is pretty important. A mistake can wipe out your tip, though an effective apology can restore it.
How much you put into getting things right depends on how important customer satisfaction is to you. Do remember that what you do will be observed with interest by other customers and that they all will likely relay the story to a number of potential other customers. In this way, if you get it right, it can actually lead to more business coming your way.