Anju and Kafei
Hey listen: For more details and other variants, check out the Tips and Tricks.
Day 1, from 10am, go to the Town Hall in East Clock Town, open the door on the right and talk to Madame Aroma. Accept to find her son Kafei and you will receive Kafei's Mask.
Day 1, at 2pm, be in the Stock Pot Inn (same district) when the postman arrives and delivers a letter to Anju. After his departure put on your Kafei's Mask and talk to Anju. She makes an appointment with you in the kitchen at 11:30pm.
Day 1, around midnight, be in the inn's kitchen. Talk to Anju and she will give you the Letter to Kafei. Drop it in a mailbox before the dawn of the second day.
Day 2, around 2pm, go to the Clock Town Laundry Pool. At 3pm, the postman arrives and delivers a letter to Kafei. Open the door from where Kafei exited, get in and talk to him when he comes back. He informs you that his Wedding Mask has been robbed in the northern district and entrusts you the Pendant of Memories. Talk to him once more, climb on the crate and take a look into the Bazar, which is in the western district.
Day 2, around 5pm, go back to the inn, talk to Anju and give her the pendant.
N64: Day 3, at 1pm, be at Kafei's.
3DS: Day 3, from 6am, go to Kafei's place.
Talk to the Curiosity Shop shopkeeper who's got a message from Kafei. He gives you the Keaton Mask, along with the Special Delivery to Mama to be delivered quickly. Then he tells you about a regular customer of his shop: a thief named Sakon.
You now have the possibility to deliver this letter to get either a bottle or a mask. Day 3, after having delivered the letter or straight away, soar to Ikana Canyon. Jump into the river, climb on the river's bank and if you have it, put on the Stone Mask if you should happen to meet Sakon who arrives around 6pm. Go to the far east, go through the tunnel and stand behind the two rocks. Kafei is hiding there, take off your mask and talk to him.
Day 3, at 7pm, Sakon's Hideout door opens and Kafei is heading there so follow him. Open the door. Now Kafei and you team up, you will be Link and Kafei in turn. In the hideout, step on blue switches to unlock doors, step on yellow switches to slow the conveyer belt down, but NEVER stand on red switches since they accelerate the conveyer belt. Kafei triggers the security system and the Sun's Mask gets away on the conveyer belt, you better be quick before it falls down through a hole.
Link: Step on the blue switch, Kafei: Go in the next room, push the left block on the blue switch, Link: Get in the room through the other door and get rid of the Deku Baba, Kafei: Open the door, walk around the red switches, step on one or two yellow switches and step on the blue switch, Link: Get in and take care of the two Deku Babas, Kafei: Go to the next room, push the block most to the left to the bottom, step on the yellow switch, pull the central block on this switch, push the central-bottom block to the bottom on the blue switch, Link: Get in and get rid of the wolf, Kafei: Push the block being in front of the door to get through, then stand on the blue stab, Link: Get in and step on the blue stab so Kafei can retrieve his Sun's Mask. Then, Kafei flies away.
Day 3, at midnight, soar to Clock Town and enter the inn. Get into the room for staff only which is located on the second floor (first door) and wait for Kafei with Anju. At about 4:30am, Kafei will arrive meeting his fiancée. They will fall into each other's arms, exchange their masks and become a couple. As a witness you will receive the Couple's Mask! Play the Song of Time to return to the dawn of the first day.
Day 1, from 10am, go back to the Town Hall, get in the left room and talk with the guards. Then wear the Couple's Mask and talk to the mayor who will thank you for helping putting an end to the meeting and reward you with a Piece of Heart. If you got the Postman's Hat, wear it and look into a mailbox to find another Piece of Heart.