Those are closing formulas for formal letters. I'll copy below what I wrote a couple of days ago in another thread .
The 'traditional' rule (for formal letters) is as follows:
If you don't know the name of the recipient, start with "Dear Sir or Madam" and end with "Yours faithfully"
If you know the name of the recipient, start with "Dear Mr (or Mrs,or Miss) + surname" (eg. Dear Miss Jones) and end with "Yours sincerely".
That said, use these only if you're writing a formal letter as a part of an ESL exam.
Yours faithfully seems to be no longer in fashion.
To the best of my knowledge, you use "yours sincerely" in case of knowing the name of the individual whom you write. In other words when you begin your letter with "Dear XY", so a specific person is adressed.
However, "yours faithfully" is used when you have no particular name in mind, and so you beging your letter like "Dear Sir or Madam".
Truth to be told it is quite commonly misused, so do not worry if you had used it wrongly, most of the people have no clue about the difference.
Hopefully, I could help.