A letter of resignation was a document that formally announced one's intended resignation to their commanding officer.
In 2371, while apologizing to his wife for his jealous outburst brought on by Lwaxana Troi's case of Zanthi fever, Miles O'Brien informed her that he'd already left his letter of resignation on Benjamin Sisko's desk, planning to go wherever she wanted as long as they stayed together. She considered it but later told him that while she appreciated the offer, he'd better retrieve the letter before the captain saw it. (DS9: "Fascination")