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Depending on how formal your email needs to be, you may want to start with dear as you would a regular letter. In most circumstances though simply addressing it directly to the person is fine. In a conservative industry you should probably start using a formal greeting including Miss, Mr., Mrs. or Ms. In a more laid back environment though, using just a first name is fine. This sets the tone as being direct but still says that you are a peer of the person to whom you are writing. That being said, if you are addressing your email to a executive officer, such as the CEO of a company, formality is probably more appropriate.
When you are emailing someone for the first time or about a new topic, you need to provide all of the necessary information so that someone can read your email and quickly make a decision about what to do with it. It may seem like a good idea to leave them wanting to know more but that's not appropriate in a fast-paced world where time is precious. If someone doesn't have all of the information they need to make a decision, they likely won't go out of their to find it and your email is more likely to be ignored (especially if you are emailing in regards to a job or internship). Your email should, in a short and concise manner- answer the key question words: who, what, where (if appropriate), how, and why.
Spelling and Grammar
Use proper spelling and grammar and avoid any abbreviations, slang, or online shortcuts you might use with friends or on Twitter, such as idk.
Follow Their Lead
You should be at least just as formal as the person to whom you are writing. That means if they address you as Miss or Mr, you should do the same. If you write to them and address them formally but they respond by signing the email with just their first name, it is okay to reply by using just their first name. Similarly, use their degree of formality to set the tone for your word choices. Also, it can be common when emailing to cut out salutations all together and just get directly to the message. Don't do that unless the other person does it first.
Putting it Together
So, let's put together an example. Let's say you heard about an internship opportunity through your professor John Do at a company Chicago called ABC Entertainment that books regional tours for indie artists. You go to school in Boston and study music business, let's say at Berklee and you're a sophomore, but are from Chicago and will be there for the summer. You have some related experience working at a local venue in Boston and you took a class on booking this year that you really liked. Your professor instructed you to send your resume via email to Jane Doe. You email should address the following:
Who are you?: A student at Berklee who is studying music business
What exactly is the purpose of your email?: You want to intern with them
Where?: This is relevant here because you're applying for an internship in Chicago but stating that you currently attend school in Boston. Let them know that you'll be in Chicago.
Why do you want to work there and why are you a good candidate?: You discovered an interest in booking, have related experience, and you like working with indie artists.
How did you find out about this internship and get this person's contact info?: Your professor John Do recommended it and told you to contact her.
Jane, (you can make this more formal if you feel awkward just using their first name)
I'm contacting you in regards to the summer internship with ABC Entertainment. My professor John Do recommended I reach out to you. I am sophomore at Berklee studying music business and will be returning to Chicago, where I am from, for the summer. I recently discovered a passion for booking when I took a course on it and I have spent the past year interning at a local venue here in Boston. I would love the opportunity to learn more about booking by working with you and I especially enjoy that your work focuses on indie artists. Please find my resume attached and feel free to call me at 555-555-5555. I look forwarding to discussing this opportunity with you.