This page explains how to get all six empty bottles that are available in Majora's Mask.
Reward for Rescuing Koume
To get the first bottle, go to the Southern Swamp that is south of Clock Town. Use the Deku Mask to hop across the lilypads and reach the Magic Hags' Potion Shop. Go into the Woods of Mystery behind the Potion Shop and follow the monkey to find Koume. Talk to her until she asks for something to revive her, then go back to the potion shop and talk to Kotake. She gives you a free red potion in a new bottle.
Reward for Winning the Goron Race
Defeat Goht to make it become springtime in the mountains. Go to the cave in the north wall of the Goron Village and talk to the Goron there to get a Powder Keg. Use it to blast the boulder in front of the Goron Racetrack. Afterward, win the Goron Race. If it's the first time you won, you get Gold Dust in a new bottle.
Reward for Saving the Cows
Before 2AM on the first day, buy a Powder Keg and take it to Milk Road (southwest of Clock Town). Use the keg on the big boulder and go into the ranch. At 2:30AM on the first day, help Romani prevent "them" from stealing the cows and she will give you a bottle.
Reward for Giving Madame Aroma the Special Delivery
Do the Anju and Kafei Quest. Give Madame Aroma the Special Delivery in person to get the bottle.
Reward for Winning the Beaver Race
In Great Bay Coast, go to the southeast area to reach the second part of the coast. Go back to the waterfall in the back and use the hookshot to go to the top of the waterfall. Go into the cave there. Put on the Zora Mask and press B to sink to the bottom of the lake. Talk to the beaver, then win the race to get an empty bottle.
Reward for Helping Dampé
Go to Ikana Canyon (east of Clock Town) and go to the graveyard to the left of the cliff that the guy is sitting on. Find the big skeleton in the back of the graveyard and play Sonata of Awakening to wake it up. Shoot it with arrows then hit it with your sword until you defeat it. Then use the Hookshot to reach the chest, which contains the Captain's Mask. Then on the night of the final day, wear the Captain's Hat and go to Ikana Graveyard. Talk to the three Stalchildren and go into the grave. Help Dampé find the correct dirt piles to dig, then kill the Big Poe and open the chest to get a bottle.