Is it "Yours faithfully" or "Yours sincerely"?

This is called "complimentary close".

As reported by Oxford Handbook of Commercial Correspondence:

• If the letter begins with Dear Sir, Dear Sirs, Dear Madam, or Dear Sir/Madam, the COMPLIMENTARY CLOSE should be "Yours faithfully".

• If the letter begins with a personal name, e.g. Dear Mr James, Dear Mrs Robinson, or Dear Ms Jasmin, it should be "Yours sincerely".

• A letter to someone you know well may close with the more informal "Best wishes".

Note that the Americans tend to close even formal letters with Yours truly or Truly yours, which is unusual in the UK in commercial correspondence.

Avoid closing old-fashioned phrases, e.g. We remain yours faithfully, Respectfully yours.

I've been taught the following distinction:

• Use "Yours sincerely" when you know the person you are addressing, i.e. Mr. Smith.
• Use "Yours faithfully" when you are starting your letter with Dear Sir/Madam, or a similar construction.

That being said, it has been my experience that these are used less and less, especially in electronic communications. I would still prefer them in dead-tree letters, but only in the most formal of circumstances (probably when invited to a cup of tea by the Queen of England...).

I usually just write "Sincerely,". I understand it to be a contraction of "I am yours sincerely" or "I am yours faithfully". If I used it, I'd probably invert it to "Sincerely yours," or "Faithfully yours,". These statements are typically reserved for love letters or other personal correspondence, although faithful could technically describe a business relationship.

Best,

The Raven

The modern era does not routinely recognize the "complimentary close" as such, and its use is becoming rather quaint.