How you should introduce yourself in an interview?

First impressions are important, so take some time to hone your introduction, and then tweak it depending upon the audience. This question is often asked in interviews, so prepare for it.
Nailing the introduction will break the ice and improve your chances of moving to the next steps. I’ve interviewed hundreds of people over the years, and often I can tell in the first few minutes if I’m interested in hiring a person.
Your main goals in the introduction are fairly simple. You need to show that you:

  • Have the ability and desire to do the job
  • Will fit into the company culture, and be good to work with
  • Won’t quit shortly after being hired

First off, you want to keep your introduction positive and simple. Don’t ramble on for too long. For instance, highlight aspects of your career, interests, accomplishments, education, and / or hobbies that match that of the company and interviewer.
Focus on what is directly relevant to the job you are interviewing for and for the person interviewing you. Keep in mind an interview is generally not about how smart you are, or how great of a person you are. It is about your fit for a specific job.
Most interviewers don’t have long attention spans, so use this time wisely. Don’t assume that your interviewer has read your resume. However, don’t simply recite your resume verbatim to your interviewer. Sadly many interviewers are not that well prepared, and don’t let that sidetrack you.
Also, keep in mind who is interviewing you. If it’s an introductory HR interview, keep things high-level. Even if you are interviewing for a highly technical job, chances are the recruiter will leave that part of the interviewing process to the subject matter expert.
If you are having trouble with this question, re-frame it to: “Why are you are a great fit for this position?”
This is your time to craft your message, so don’t bring up anything negative! Even if you are reaching for the job, give an introduction that shows why you deserve it. Never talk yourself out of a job or hedge in the first few minutes.
For example:

  • If you are a recent college graduate, highlight internships, hobbies, or college projects that relate to the job.
  • If you are making a career switch, highlight your positive attributes and transferable skills.
  • If you are looking to make the move to management, highlight specific examples of how you have motivated and led teams.

For a few examples of other things to avoid: see Mira Zaslove’s answer to What are some of the biggest red flags in an interviewee?
If you have been referred to the company by a current employee, it is appropriate to mention that they got you excited about the position. Also, if you know anyone else at the company, and feel that they will give you a positive recommendation, let the interviewer know.
Finally, if you are feeling uncomfortable and having a hard time reading what the interviewer wants, it’s fair to ask them. For instance, say something along the lines of: “I have 10 years of work experience, and want to make sure I best answer your questions. Can you tell me what you would like me to focus on?”