This is an extremely common job interview question, and the purpose behind it is two-fold. One is to get you talking and warm you up a bit. The second, primary objective is to find out about how "big" you are early in the conversation. What you consider to be your greatest achievement or greatest accomplishment will tell the interviewer much about what you value in your career, the business impact of your career, and deep cultural characteristics.
For example: You are interviewing with a company that is culturally bent toward financial analysis, metrics, and tangible measurement of success. You describe the greatest accomplishment of your career, and the cultural follow-up question comes:
"Tell me why this was a great accomplishment."
If your answer does not include the exact financial impact on the company, but instead something squishy like "It made the boss very happy," you will never get an offer from that company. They are looking for people who can quantify.
Therefore, always be prepared with several detailed examples of key accomplishments in your career. Rehearse your answer to this question carefully. Make sure you understand why this was a great accomplishment. Know why this was important to the business in measurable terms, not just why it was important to you.
Lastly, if they don't ask this question, make sure you find a way to work your examples of greatest accomplishments into the interview. Try to work them into other questions if possible, but tell the interviewer what kind of difference you have made.
"I’d have to say that my greatest accomplishment was a couple of years ago (specific example), when I had to develop a whole new customer relationship system for ABC Company. I had only 9 months to complete a 15 month project, so I crashed the timeline wherever I could, influenced other departments to loan me some staff, and got approval to bring in a couple of consultants who specialized to get it done on time. We finished the project on time and under budget. In spending some money to crash the project and complete it at 9 months, we created about $85 Million in new sales for the company that we would otherwise have missed (business impact). I’m very proud of that accomplishment."
More sample interview questions than you can imagine are in the Interview Questions and Answers Database. This resource is affordable for almost anyone and is highly recommended.