The post box is a yellow box for sending mail, found outside Teimo's Shop right next to the door. It's used to send orders for aftermarket parts from the part catalogue in the player's house. To send a letter, grab the letter (left click) and move it above the post box. The flap will open up on the top. Simply drop the envelope inside the post box (left click) and you're done. It will disappear when falling in or moved around the mailbox, indicating that the envelope has been sent.
The player can drop off the envelope at any time of the day, even if the shop is closed. Simply go home and wait a few in-game days. You will get a call on the telephone, and Teimo will inform you that your post order is ready to be picked up. Simply drive to the store, pay for the order, pick your items up from the side of the store, and drive back home.