0th Day is a glitch which changes the time to 5:59 AM on Day 0. Because it is not supposed to be day 0 during normal gameplay, this glitch has various effects which are related to which day it is. These effects will not work on the Japanese version because you cannot save at owl statues.
This glitch has various effects in different areas, which can include vanishing actors. Most notably, when used in Romani Ranch, the Aliens will be defeated instantly, so you can obtain Romani Mask without shooting any aliens or having a bow.
The following video documents most of the changes which occur when you activate 0th day in different areas.
When you select "Yes" on the stored Song of Time text box, your current file's save information will be copied elsewhere. After 3 seconds pass, it will be copied back. This can be used for various tricks and glitches, because the game is partially reverted to the state it was at when you saved.
If you activate 0th day then switch masks quickly, the state restoring effect will restore which buttons were dimmed or undimmed before the switch. This lets you equip any item over the undimmed buttons and use them, as any form.
This trick notably allows you to unfreeze Goht without a bow, by changing to deku link after activating 0th day, then equipping fire arrows on an undimmed button.
The 0th Day state restoring effect can allow you to reload your previous entrance instead of a new one, or load the first entrance of the current area.
This is known as a 0th Day Warp, and there are several ways to accomplish it.
0th day was the original method of getting an item on C with a bottle in the corresponding slot. Equip swap's discovery allowed an easier and more flexible method without 0th day. This is generally used to keep item amounts between cycles, because your item counts don't get reset to 0 if there is a bottle in the corresponding item slot.
Steps to perform (0th Day Method):