"Yours sincerely": always used when you start a letter with "Dear Mr. John", "Dear Mrs. Bloggs" etc (i.e. you know and use the name of the person being addressed); "Yours faithfully", always to be used when you start the letter with "Dear Sir" or "Dear Sirs" etc. (i.e. you do not use the name of the person being addressed, usually because you don’t know it anyway); "Yours truly", more informal and shows a closer than formal relationship.