If you are particularly keen to be offered a job and wish to increase your profile and chances of receiving a job offer after attending interview, you can follow up an interview with a letter or email (and then a phone call) to reinforce your commitment and qualities for the job. The sooner the better.
Often jobs are offered to the most passionate and determined applicants, so this should be the feeling that your follow-up should try to convey, without giving the impression of desperation or crawling.
You should seek to focus your follow-up letter or email on the key performance aspects in the role that the interviewer believes are required for the successful applicant.
This type of follow-up enables you to show that you have considered and developed your thinking after the interview (a desirable attribute), and also enables you to re-emphasise your claim to the opportunity, bringing your name to the front of the interviewer's mind again. A good follow-up letter or email also enables you to demonstrate that you are persistent, professional, interested, possess relevant capabilities, recognise what the requirements and priorities are, are keen, and can sell yourself in a determined manner, that probably the other applicants will not do.
Interviewers also respond well to applicants who really like the company, especially if your reasons coincide with the reasons that the interviewer likes the company too, so it can help if your follow-up 'resonates' with the feelings of the interviewer about what is required for the role.
From the interviewer's perspective - if you are an interviewer or decision-maker who receives a good follow-up letter from an enthusiastic interviewee - I recommend you give the applicant extra credit and consideration. They are demonstrating many of the most relevant qualities that you are seeking.