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

When should one sign a letter with "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...).