Save data is, for most people, the most important thing that exists on the 3DS. While games can be re-downloaded from the Nintendo eShop and homebrew applications can be re-downloaded from the open Internet, save files are one of the user-generated things that cannot always be replicated in the event of catastrophic failure. It is therefore natural for those who have the means to want to back up their save data in some way.

The Nintendo 3DS stores save data in the following ways:

Cartridges usually have a place on the cartridge itself to save their data. This means that a cartridge's save data will remain when the cartridge is used on another 3DS.
- There are a few exceptions to this. Such games will save as "extdata" (extra data) and will follow the same rules as digital games.
Digital games save their data into a packed, encrypted format within the Nintendo 3DS folder. This means that in order to use your save data on another 3DS, you would need to unpack and decrypt it.
Cartridges and digital games save less important data (for example, Pokémon battle videos and Super Smash Bros. replays) as "extdata" in a packed, encrypted format within the Nintendo 3DS folder. Like the save data for digital games, you would need to unpack and decrypt it in order to import it to another 3DS.

With this in mind, there are many ways to export your save files into a format that:

Can be copied to many locations (e.g. computer, online storage, external hard drive)
Can be used for save file editing (e.g. PKHeX, Marc Robledo's ACNL Save Editor)
Can be used on another 3DS, or on Citra

The instructions here assume that you have custom firmware. If your console is unmodified or only has homebrew access through a game like Steel Diver: Sub Wars, you should install custom firmware first.

Cartridges

Since your cartridge save data is (generally) on the cartridge, you can make your save data backups on any 3DS using homebrew applications.

There are two save managers commonly used on the 3DS: JKSM and Checkpoint. If you followed our guide to install custom firmware, you will probably already have Checkpoint installed. If not, you can get it from here. Once you have it:

Open Checkpoint (the flag icon) through your preferred method
Select the game you want to make a save file backup of
- Cartridges will generally be the first title displayed on the list and will be labelled as "Media Type: Cartridge" on the bottom screen
- The game will not show up in Checkpoint if a save file has not been created in the game
- If you are having issues getting the game to show up, try holding B to refresh Checkpoint
Press the L button to back up the save file for the game
- The save file name can be anything, but setting it as the date makes it easy to identify when the save file was created

You're done! The save file will be output to /3ds/Checkpoint/saves/<game name>/<folder name from earlier>.

Digital games and extdata

If your 3DS is working

If your 3DS is currently reasonably functional (as in, you can reasonably navigate through all the menus), you can make your save data backups on the 3DS itself using homebrew applications.

There are two save managers commonly used on the 3DS: JKSM and Checkpoint. If you followed our guide to install custom firmware, you will probably already have Checkpoint installed. If not, you can get it from here. Once you have it:

Open Checkpoint (the flag icon) through your preferred method
Select the game you want to make a save file or extdata backup of
- The game will not show up in Checkpoint if a save file has not been created in the game
- If you are having issues getting the game to show up, try holding B to refresh Checkpoint
- You can press X to switch to backing up extdata
Press the L button to back up the save file or extdata for the game
- The save file name can be anything, but setting it as the date makes it easy to identify when the save file was created

You're done! The save file will be output to one of the following locations:

/3ds/Checkpoint/saves/<game name>/<folder name from earlier> if it is a save file
/3ds/Checkpoint/extdata/<game name>/<folder name from earlier> if it is extdata

If your 3DS isn't working

If your 3DS is currently nonfunctional (as in, you can't boot to home menu and the issue cannot be fixed), you may be able to make your save data backups depending on the files you have. You'll need any of the following:

Your NAND backup (e.g. 220101_AA000000000_sysnand_000.bin)
essentials.exefs
movable.sed

If you don't have any of these files on hand, you may still be able to get your encryption key if:

It is cached on Bruteforce Movable (enter your Friend Code and ID0 combination to check; if it skips to Step 4 then you're good to go)
You have (physical or non-physical) access to a 3DS that has done a two-way friend exchange to your broken console at some point (see here for instructions)
You have a Mii QR code created with your broken console at some point (see here for instructions)
Your broken console can still access GodMode9, whether through a boot9strap installation or an ntrboot cartridge

Once you have your encryption key on hand, you can either:

Import the encryption key to a working console (from which you can use Checkpoint, as above), or
Create unpacked, decrypted save data with the help of some computer programs

Import all SD card data to a working console

With this method, you lose access to all games/savedata/themes/DSiWare that was installed on the working console beforehand. If you have a large amount of preexisting data you would like to keep, choose another method.

If you perform this method, you must restore your console's original movable.sed before uninstalling custom firmware, or else performing the uninstall will brick your console.

A guide to directly import all SD card data from a different movable can be found here.

Use a computer to unpack and decrypt save files (easy)

(ninfs + 3ds-save-tool)

Use a computer to unpack and decrypt save files (hard)

(3ds-save-tool + secrets.py)

Things you can't or shouldn't do

3DS Flashcart stuff

3DS flashcarts are not advised for several reasons, and as such will not be recommended here either.

If you use such a flashcart, the save files can likely be imported into Checkpoint and used through there.

Powersaves stuff

Official Save Data Backup feature

This feature is not 'bad' per se, but Checkpoint is a direct upgrade to its functionality.
As such, there is no reason to use the Save Data Backup app with CFW installed.