If you submit your resume but don't hear anything from the potential employer, writing a follow-up letter is a great idea, and it may help you land the job. (A follow-up letter isn't the same as a thank you letter, which you should have already sent.)
Sometimes, submitting your resume can seem like dropping it in a bottle and throwing it out to sea. A week goes by, and then another, but you never hear anything from the company. You wonder if they have even received your resume, or if it's still floating around out there.
It's important to remember that the people who look at your resume can be very busy. If you submit it to a large company, the Human Resources department may be handling many resumes for many positions all at once, and the bureaucracy of hiring personnel can slow down the process even more.
If you submit it to a small company, the person in charge of hiring might also be in charge of other aspects of the company and may be tied up with the day-to-day business of keeping the company going.
Either way, if nearly three weeks have gone by and you haven't heard anything, a follow-up letter is certainly a good idea. If you have been rejected for the position, a follow-up letter will make them tell you that instead of leaving you guessing.
The letter should do the following:
Remind the employer that you're interested in the job
Express your continued interest in the position
Give your contact information
Including your contact information is important because employers can lose or misplace paperwork just like anyone else (but don't mention that in the letter!). If he or she has lost your resume, or if your resume never got there in the first place, having this contact information in hand gives that employer easy and immediate access to you, which, after all, is what you want.
Sending a follow-up letter can have another positive side effect. Although it might not get your resume moved to the top of the list, a follow-up letter showing your continued interest sets you apart from the other applicants and can move your name to the top of the employer's mind.