I am a beginner designer, and I charge $15/hr. Is it fair to charge for my time when I am correcting a mistake that I made?

Erm, no, of course not. If you made the mistake then you should correct it free of charge and not expect your client to pay for your ineptitude.

Noops, you should pay $15 to client if you MADE mistake;). Well if you are freelancer designer, make your set of rules and give it to client before starting the project e.g. what is the bug fixing period after deployment (may be 2-7 days, free of cost) if client doesn't contact you in this period then I don't think so that it is unfair to charge amount since in software we cannot afford free of cost life time support. Because designer mistake is different from programmer mistake, if you did any implementation mistake, it may ruin client whole business but designer work can be judged by any person (IT or non-IT) so client should contact within support time to fix bugs. Well, this was my way of work when I used to do the freelance, you should think according to your skill and experience.