If you just met a girl you like, had a date with her or otherwise interacted with her and it went badly, you don't need to worry. Girls are people too, and most of them will probably be happy to give you a second chance. If you want a second chance, you just need to be honest and forthright and ask for one. She may not necessarily say yes, but if she says no, you are ultimately better off as there are many women out there who would love to give you a second chance.
Acknowledge the fact that you made mistakes or a poor impression during your first chance with the girl. This shows that you are both humble enough to apologize and self-aware enough to understand that you did not make a very good impression, which she will likely appreciate.
Move onto another subject. If you apologize too much, you will seem insecure and self-obsessed. So, once you've apologized, start talking about something else.
Make a joke. It doesn't have to be particularly funny or inspired, but the effort of making a joke shows that you don't take yourself or your situation too seriously.
Keep an eye on yourself during the second, and any future, chances. This doesn't mean you should act like someone else, but you should try not to offend her again. So, just be nice and be yourself from now on.
Move on if she doesn't give you a second chance. There are plenty of women who would love to give you a first, second and third chance; if this girl isn't interested, then you need to look for someone else.