The interviewer wants to see if you the have self-awareness required to understand your weaknesses. Knowing the areas in which you can improve, is a sign of maturity. Share something that you have taken the time to reflect on. Maybe you made a mistake by moving too quickly through a project, and not paying attention to detail. Whatever it may be, show your ability to adapt and change from the mistakes or behaviors that didn't serve you in the past.
"I tend to be a people pleaser. In the past, I have said 'yes' before looking at my schedule, and I have overcommitted myself. Now whenever my boss asks me to do something, I let him know I need to check to make sure I have the bandwidth. This practice has saved me from taking on too much and overworking myself."
Entry Level answer example
"My greatest weakness is caring too much about the small details. It can be easy to care a great deal about the minute details. I have worked on this by ensuring that I remind myself of the big picture."
Experienced answer example
"My greatest weakness is in taking control when leadership opportunities present themselves. I am not a naturally outgoing or gregarious personality. I have taken the opportunity to sign up for a free leadership skills workshop at the local library. I hope this will help me to hone my leadership skills further."