an official letter: a written message sent from one party to another, usually carrying royal or administrative authority
This word names a letter in the formal sense: a written dispatch carried from a sender to a recipient, most often between people who hold power. Kings send it to their governors, officials send it back and forth, and a national decree is published through it. It is not a word for writing in general but for the physical document that carries an official message.
All ten uses fall in the later books of the Old Testament, and they share one picture. Hezekiah sends letters summoning the nation to keep the Passover; the Persian king grants Nehemiah letters of safe passage and supply; Nehemiah's enemies send letters meant to frighten and discredit him; and the festival of Purim is established and confirmed by letter. In every case the letter is an instrument of authority, not a private note.
Because the word always points to correspondence with weight behind it, it sits close to the broader word for any written document, סֵפֶר, which can mean a book, a scroll, a deed, or a letter. This word is the narrower of the two: it is only ever a letter.