I assume this question has come up on a job application although it is a useful question for any career changer to ask themselves.
Are you a job seeker? As far as a job application is concerned, it is important to think about what the recruiter is looking for. Job applications are all about showing that you meet the needs of the employer so everything you say about yourself should be written with their needs in the front of your mind.
So if they are looking for someone with good customer service skills, then you could talk about how proud you were to give excellent service to clients in a previous job. Give and example of a situation that led to a very happy customer.
Similarly if they are looking for a methodical approach and attention to detail, find an example of where you demonstrated this and explain that you were pleased with your efforts and perhaps were praised by your boss for your work.
By all means stop and think about the moments that you took a personal pride in. Maybe you kept the peace between 2 difficult staff members or you took on extra work above your level when someone was off sick unexpectedly. If you start with personal 'proud moments', make sure you relate them back to the job description of the role you are applying for.
Are you changing career? For career changers, this question can throw light on what really matters to you at work. Were you proud of yourself when you pushed yourself to go for a promotion, or perhaps when you rejected a promotion because you knew it was just not right for you? Maybe you achieved a high status and salary, but the things that you actually took pride in were more to do with acquiring new skills or training junior staff.
This is a great question for revealing what your priorities and values are. Recruiters will want staff whose priorities align with their own. Career changers will want to be sure that new career ideas they are considering will align with their own values.