The first question you will be asked in any interview, networking event, job fair, happy hour, professional event, etc. is "tell me about yourself" (or a similar variant like what do you do or to introduce yourself). So you absolutely need to have a succinct and impressive introduction, also known as an elevator pitch, ready to go at all times -- especially if you're looking for a job.
Assume they know nothing about you
Just because you got an interview doesn't mean you should assume the hiring manager (or committee) memorized your cover letter and resume. It's more likely they were impressed at the time but have been in a series of back to back interviews and, if they were lucky, they had a chance to glance at your application again before you arrive. So do not introduce yourself simply by saying your name and current job title. Use this as an opportunity to tell your story and work in a few impressive details to set the right tone.
Who you are
Obviously this includes your name. But, this is where you talk about your past and your experience including something that clearly and quickly establishes your credentials. For example, if you're just starting out you might say I'm majoring in music business and have interned at a record label and a venue (include the names if you think people will know them). Eventually, you might say you're a GRAMMY-winning musician (wishful thinking?).
What you do
Talk about what you're good at and how you can help the company, be a little bit specific. Let's go back to the example of the record label intern. If you worked on social media and ran street teams for major label artists and you're applying for an entry level social media job, then say that.
Why you're a good fit (and why you want it)
Talk about the skills and experiences that make you a good candidate for this job and why you're interested in this specific company.
Anticipate and address concerns confidently
Are there any obvious gaps in your resume? Quickly and confidently address them here. For example, if you go to college in Florida but are applying for a summer internship in your home state of Minnesota then speak to that and put a positive spin on it.
Putting it all together with a (mostly) real example
I'm Katie I've worked in music marketing since 2006 with both major labels and indie artists. I've run street teams email lists, created web content to support top-selling record releases, and managed artists' social media accounts. I'm passionate about artist development and using digital tools to build a strong fan base, that's why I'd be excited to be a part of your digital marketing team when I move to LA next month.
This introduction quickly established that I have several years of experiences with different types of artists. It listed, with some degree of specificity, what I do and that I did it well enough to be successful (the part about it being top-selling). It spoke to my specific interest in that role and company (I'm assuming the fake company is interested in artist development) and it addressed the fact that while I don't currently live where the job is located I am moving there soon.