Request letters are used for many things, such as for requesting funds and grants, leaves of absence, asking for a reference and so on.
These letters have to be very well written to get a positive reply.
The essence of a request letter is to be convincing, to come up with reasons why you deserve to gain what you are requesting. Enumerate as many facts as you can to support your cause and request in such a letter, and do it shortly.
In a letter of request also make sure that you maintain a professional tone. The more formal you are, the more chances you have that you will get a positive response.
When you are writing the letter, think about the audience or the person you are writing to. Do not think about yourself. Focus on what they mean and what their organization means. Include why you have chosen them and not someone else if the request letter is asking for a donation, for instance.
Don't apologize in such a letter: for example saying Sorry to bother you is not very professional and it is also not necessary. Don't be manipulative either: these things are very transparent and they don't make you look good at all.
Dear Ms. Rooney,
I have applied to your job online for the position of engineer within your company. On the 12th of March I have had a phone interview with you, after which we agreed on a scheduled interview on the following Monday, 18th of March.
I am writing to you with the request of rescheduling the interview mentioned above. Due to health reasons I cannot be present on the date that we agreed upon. I would highly appreciate if you could postpone the date of the interview. I am eager to meet you and to become a member of your admirable group. I hope that you will find my skills and my experience just as valuable as I find your company.
Thank you for your time and I will be awaiting your call for rescheduling the interview. Looking forward to meeting you!