Safe Mode is an option built into every 3DS that allows the device to perform a System Update without booting to the HOME Menu. You can access Safe Mode by holding (Left Shoulder) + (Right Shoulder) + (D-Pad Up) + (A) on boot. If you were successful, the 3DS will show the System Update screen and allow you to update.
There is no alternative key combination, so if one of the necessary 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 generally because Nintendo is more reluctant to update Safe Mode to patch exploits (since an unmodded 3DS would have no other ways to fix a brick caused by a system update). Another outdated exploit called safehax allowed the console to load a payload in the .bin format after being loaded from Homebrew Launcher due to the Safe Mode firmware being outdated and the console being able to load SAFE_FIRM (the Safe Mode firmware) when the console was booted into normal firmware. This exploit is OUTDATED and will not work in versions higher than 11.3 since Nintendo released a fix that made the console unable to boot SAFE_FIRM after a normal boot.
Redownloading system applications
Just like a normal System Update, a Safe Mode update will redownload some system applications if they have been somehow lost, which can get you out of some tight situations.
Unbricking
Because Safe Mode is (somewhat) separate from the normal firmware, it can help save you from certain kinds of bricks (such as 3DS:Keyboard Unbrick, since the keyboard will work in Safe Mode even if it is broken in HOME Menu). That being said, the best way to avoid bricks is to keep custom firmware installed.