Fan mail was a way of fans reaching their favorite celebrity or someone of renowned status by sending them a letter.
In 2376, while visiting the Qomar homeworld, The Doctor became an icon to the Qomar people when he introduced music to their culture. The USS Voyager was overrun with fan mail from the Qomar to the Doctor. After leaving the Qomar homeworld, the Doctor received a fan mail letter from Seven of Nine to cheer him up after the Qomar replaced the Doctor with a more sophisticated hologram. (VOY: "Virtuoso")