Safe Mode, also known as SAFE_FIRM, is a part of the 3DS' stock firmware that allows the device to perform a System Update without loading the HOME Menu or most system functions. To access Safe Mode, hold L + R + (D-Pad Up) + A and then powering on - if successful, the 3DS will show the System Update screen.
There is no alternate keycombo, so if one of the needed buttons is broken, you will be unable to access Safe Mode.
Why it's useful
Exploits
The custom firmware method recommended for 11.16.0, unSAFE_MODE, uses a flaw in Safe Mode that can be used to install custom firmware. This is because Nintendo is more reluctant to update Safe Mode to patch exploits (since an unmodded 3DS would have no way to fix a brick by doing a system update if SAFE_FIRM were also bricked).
Another exploit that is now unusuable called safehax allowed the console to load a .bin payload after being loaded from Homebrew Launcher due to the Safe Mode firmware being outdated and the console being able to load SAFE_FIRM (the Safe Mode firmware) when the console was booted into normal firmware. This exploit was given a partial patch in version 11.3 and fully patched in version 11.4.
Redownloading system applications
Safe Mode reinstalls most deleted or corrupted system titles, the same as a regular System Update, which can also help if you need one of those titles to fix another issue.
Troubleshooting
Because Safe Mode is in a unique partition of the 3DS' firmware (SAFE_FIRM), it can fix crashes on boot that are caused by corrupted/deleted system titles or extremely outdated firmware, as only NATIVE_FIRM is affected by these. The best way to avoid bricks is to keep custom firmware installed, however, and most crashes not in the above list will require access to GodMode9 to fix - if you have an issue that disables access to GodMode9, you will most likely need a flashcart compatible with ntrboot to fix the issue.