This page explains how to save the game in Majora's Mask. Unlike in other Zelda games, you can't save the game whenever you want to. Instead, there is an Ocarina Song that you can play to save the game, or you can use Owl Statues that are scattered around the land.
Saving with the Song of Time
When you first start the game, you won't be able to save until you have learned the Song of Time. It will probably take at least 15 minutes of playing to reach that point. To learn the Song of Time, you need to reach Clock Town as a Deku Scrub, then bring the stray fairy to the Great Fairy fountain, pop Jim's balloon with a bubble, find the five Bombers to learn the password, get a Moon's Tear at the Observatory, give the Moon's Tear to the Business Scrub in front of the Clock Tower, then wait until midnight on the final day (or talk to a scarecrow to move time forward). After midnight on the final day, use the Business Scrub's flower to go up to the upper level of the Clock Tower, and go up the stairs there. Blow a bubble at Skull Kid and pick up your Ocarina. Then you will learn the Song of Time and will automatically save the game. For a more detailed guide, read the Beginning the Game and Clock Town page.
From then on, you can play the Song of Time whenever you want to save the game, but it will send you back to Dawn of the First Day. Read the next section to see how to save without going back to the first day.
Use Owl Statues
After you learn the Song of Time and go back to the Dawn of the First Day, talk to the Happy Mask Salesman in the Clock Tower to be transformed back into your human form.
When you can use your sword again, look for owl statues in different places around the world and hit them with your sword. For example, there is an owl statue next to the Clock Tower. Once the owl statue's wings are open, you can talk to it to save and quit your game. (Note: the owl save feature is not available in the Japanese version of the game.)
However, after you load an owl statue saved game, be sure to save the game again before quitting. The saved game at the owl statue is temporary: it can only be loaded once. After you load an owl save, you have to save the game again, either with an owl statue or the Song of Time, to avoid losing your progress. If you reset the console after loading an owl saved game, your saved game will go back to the most recent time that you used the Song of Time to save. In other words, if you don't save again after loading an owl statue saved game, you will lose all of your progress since the most recent Dawn of the First Day.