Hello, this is a common question at interviews, and by preparing ahead, you can conquer this meaningful question.
The goal of this question is to determine if you will be a good fit in the organization or department; to find out how you handle your own short-comings; and to see if you are humble (versus boastful).
Therefore you should give an example and a solution as to your strengths and weaknesses.
1) Example of a weakness: You have trouble getting places on time. Solution: You prepare the night before by having your clothes laid out and ready and have everything that you need to take the next morning to work.
2) Example of strength: You are very organized - you have a place for everything. Humble answer is: You know if you do not have a place for everything it would cause you to stress out and be very frustrated.
The employer wants to know how well you know yourself and how you handle life's ups and downs. They want your honest answer (without telling all your business) and to find out if you are a balanced person. Additionally, they want to know if you are a leader.
When you give an example, use one that the employer can identify with.
Do you get the picture? I hope this helps.