The basis for asking this question is for the employer to determine how well you handle adversity. There are two parts of the question that needs to be addressed:
This question is asked specifically to put you under pressure during the interview. They want to see how well you handle a little pressure. To pass this test you have to prepare for this question to be asked. So have an answer ready before you walk in the door.
Everyone makes mistakes. They know that. They want to know how you cope with your mistakes. Do you shift blame? Do you downplay the importance of your mistakes? Or do you take responsibility for your mistake? Do you having a coping mechanism for it?
The answer I give for this question is to tell them a short story. I work in IT so I tell them about a bad experience I had on a former job where my supervisor was not in good communication with the IT director and I was given conflicting order. I explained that I tried to resolve it with my supervisor first but he was firm that I listen to him only. Whereas I had no authority to disobey the director. Quite the conundrum and is actually a pretty common occurrence in office jobs. So I tell them my worst quality is that I don't thrive in an environment where my upper management is at odds with each other. I explain that it leads me to confusion, frustration, and frankly it makes me no longer feel like part of a team. I then go on to explain that when things like this are unclear, I simply use my best judgement and will be accountable for my decisions.
I like this answer because it sets the stage that I'm looking to be part of a cooperative environment, that I am capable of making my own decisions, and that I'm willing to be accountable for those decisions.
Whatever answer you choose, you have to remember that it needs to end in a positive story about how you are a qualified candidate.