I am looking for a few things:
Did you own the mistake. (responsibility)
What steps did you take to resolve it. (accountablity and ownership)
Are you able to laugh about it, or did you learn something from it. I would hope that you can make claim to one or the other. (personality)
Can you tell a story. Because the story is personal, it is topic where you have "mastery" and should demonstrate how you communicate. (communication)
An understanding of your professional background - most of these stories have other "players" in them that help move them along and tell me who you have/had to interact with in the past. (experience)
Remember I give my own example that not only hits these points but probably illicit a laugh our two out of the candidate, and remind them that I am not any different from them. I tend to be able to have more conversational interviews after this question, and get better responses and answers from candidates.