Instructions
Purchase stationary from Tom Nook's store to write letters. You can also find stationary in the town's lost and found, receive it as a gift from your neighbors or find it in the town dump.
Open your menu and select the stationary. Choose "Write a Letter" from the sub-menu and select which neighbor you want to write a letter to.
Type a short message using the on-screen keypad. The message must have proper spelling for the letter to be effective, otherwise the neighbor will not know what you meant and will not send a gift. Use positive words like friend, please, fruit, present or the name of your town.
Save the letter and drag an item onto the envelope in your inventory to attach a gift. It does not matter what you give as a gift, so pick a free item like a piece of fruit or seashell.
Take the letter to the post office and speak to the pelican behind the counter to have the letter mailed. You should receive a response to your letter with a gift attached within a few days.