If you are writing to someone whose name and title you do not know, use the greeting Dear Sir or Madam, and the ending Yours faithfully, signing yourself with your initials and surname.
If you are writing to a named person, address them as Dear Mr/Mrs/Miss/Ms, and end Yours sincerely, followed by your first name and surname.
If you have met them or spoken to them by phone, or otherwise feel that you have some acquaintance with them, address them by their first name and sign yourself Yours sincerely, using your first name.
SOURCE: AskOxford