In 1928 H. W. Fowler listed these phrases and their uses:
Yours faithfully (to unknown person on business)
Yours truly (to slight acquaintance)
Yours very truly (ceremonious but cordial)
Yours sincerely (in invitations and friendly but not intimate letters)
With slight variations between British and American usage, these forms are still in use.
If you don’t know the name of the recipient…
Yours faithfully is British usage. It is used when the recipient is not addressed by name, as in a letter with a “Dear Sir” salutation. I have never seen it in correspondence between Americans. That’s not to say it won’t catch on. I’ve come across letter-writing guides on the web that imply that it is standard American usage.
Yours truly is the American equivalent of “yours faithfully” that I was taught by my American business teachers. When I begin a letter “Dear Sir,” I close it with “Yours truly.”
When you do know the name of the recipient…
Yours sincerely is also British. Americans tend to reverse the order and write Sincerely yours.
When I worked in England, I was told that to write Sincerely without the Yours was very bad form. Now, of course, Sincerely is a common and acceptable close for American business letters.