The mailman is the Gravity Falls mail carrier.
His first appearance is in "Tourist Trapped," when Soos mentions that he frequently notices odd things in Gravity Falls, like the mailman, whom he believes to be a werewolf.
Later, in Dipper vs. Manliness the scout lady mentions that she is looking for the mailman to deliver some letters she is carrying.
In Weirdmageddon 2: Escape From Reality and Weirdmageddon 3: Take Back The Falls, the mailman is one of the people in Bill's Throne.
He is covered with reddish-brown hair, wears a blue mailman uniform with a hat, and holds a mail bag. His chest hair puffs out of his shirt, he has bushy eyebrows, a beard, and a mustache.