The Battle Save feature is an option present in certain games throughout the Fire Emblem series. It is a feature that allows the player to save the game in the middle of a chapter. It is different from the Bookmark feature in that Battle Saves are a permanent save file, while Bookmarks are merely a temporary save file that is erased upon loading.
The feature debuted in Genealogy of the Holy War, where the player is allowed to make Battle Saves at the beginning of a new turn. As chapters in this game are quite long, this is a helpful and sometimes necessary feature to make use of.
Following Genealogy, Battle Saves did not return until Radiant Dawn, where they then became a regular feature of the series from that point on. In Radiant Dawn, Battle Saves can now be made at any point during the Player Phase, except in Hard Mode, where they are disabled and replaced by the "Bookmark" feature (also known as "Suspend"). The Battle Save feature was also completely absent from the original Japanese release of Radiant Dawn.
In Shadow Dragon and New Mystery of the Emblem, Battle Saves are present in a much more limited form. Here, Battle Saves can only be made at "save points", glowing circles that are present on the map. You must move a unit onto a save point and use the "save" command, which allows you to make a single Battle Save and concludes the unit's turn. Save points disappear after a single use, and cannot be used again unless you restart the chapter, so they must be used wisely. Some chapters feature multiple save points.
Battle Saves are very useful in the event that the player loses a unit, since they can reload the Battle Save until the outcome becomes positive, rather than restart from the beginning of the chapter. Some players would also make a Battle Save right before a level up occurred, and then reload the Battle Save if the level up does not produce good stat increases, and try again in hopes of better stats. This can happen because the Battle Save does not reuse the same random numbers but generates new ones with each reloaded save. These manipulations made it much easier for characters to cap their stats, or just generally get decent level ups. However, this cannot be done in Fates, as the game will remember the last few random numbers that were generated, so you must do things differently if you wish to change the outcome of a battle or a level up.