Letters of interest (LOI) - or what some applicants jokingly call "love letters" - can be used in several types of situations:
1) Residency applicants who want to make an impression post-interview before program directors (PD) submit their rank lists
2) Medical school applicants who have been interviewed but have not yet been accepted or rejected
3) Medical school applicants who have not yet been invited to interview (It's a little late in the cycle now for residency applicants, but LOIs can be used in this way in October through December for residency.)
4) Medical school applicants who have been wait listed
When writing your letters of interest, you should have the following goals:
1) Restating your strong interest in the institution
2) Positioning yourself as a distinctive candidate who can contribute fully to the institution
The biggest errors I see in LOIs are:
1) Too much content about a specific institution's advantages. There is no reason to spend a paragraph or more telling a PD or medical school admissions member what makes their institutions special. Using your precious space this way is an opportunity cost, keeping you from fully showcasing what makes you a compelling applicant.
2) Writing about how the program or school will help you. The focus should be on how you will be a contributor to their institution. (It's the "What have you done for me lately?" principle.)
Finally, let me say that I am coming to think that the term "letter of interest" might be better than "letter of intent" because, generally, I don't recommend you let institutions know what your intent is since it might change. Note that you can create one general letter that you modify and send to different institutions with different goals (i.e. requesting an interview or demonstrating your interest post-interview). You can also modify the letter and send it to your top choices because, if you are not making commitments, then you can honestly use it for multiple institutions.