BootMii is a modification of boot2, which loads the system menu when booting the Wii. As boot2 is normally not updated through a wii system update (which was proven wrong with the 4.2 update!), this is the best brick protection you can get. Unfortunately, Nintendo already knew about it before it was even released, so they updated boot1 (which loads boot2) on newer Wii's. This is why BootMii can't be installed as Boot2 on those Wii's.
One of the features BootMii provides is the ability to create a NAND backup. The NAND is the Wii's internal memory. BootMii has the ability to dump this memory to the SD card (as the file NAND.bin). This backup can also be restored later on if needed, providing an additional brick protection.
IT IS HIGHLY RECOMMENDED THAT YOU CREATE A NAND BACKUP AS SOON AS POSSIBLE AFTER INSTALLING BOOTMII.
Using BootMii is not difficult. Every time you boot the Wii with the prepared SD card inserted, you will see the BootMii menu.
If you only installed BootMii as IOS, you need to launch BootMii through the Homebrew Channel instead. In the Homebrew Channel, press home to access the menu, and choose "Launch BootMii".
You can navigate through the menu with a GameCube controller or the buttons on the Wii itself (Power = right, Reset = OK). The WiiMote will not work.
To make your Wii more brick-proof, make a backup of the NAND before making major changes to the Wii memory. The NAND backup function can be found under the 4th button, then the 1st button (the highlighted button in the below screenshot).
The first option is NAND backup, the second is NAND restore
If you get any Bad blocks during the Backup, don't panic! This is completely normal. Once it completes making the backup and verifying it, put your SD card in the computer and Copy the NAND.bin to a safe place.
Also copy over the keys.bin file. These are the unique keys of your Wii!
If you need to restore your NAND backup, place it (the nand.bin) back in the root of the SD card and choose the restore NAND option (to the right of the backup NAND button).
A useful program you can use is BootMiiSwitch. It allows you to turn BootMii on or off from inside the Homebrew Channel by renaming the "bootmii" folder on your SD Card to "bootyou". When you run the app again it'll rename "bootyou" to "bootmii" once again. If you have both a "bootyou" and "bootmii" folder then it'll swap the two.
Another useful program that fixes the "Backup from another Wii" error when recovering your NAND is called Bootmii-KEYFIX.exe. Just run in the same folder as your NAND.bin and keys.bin
How it works
The booting procedure of the Wii goes something like this:
Boot0 -> Boot1 -> Boot2 -> Wii Menu
We can see that there are various "booting" steps that happen before the Wii Menu is executed (which is when the health screen appears).
Boot0 is on read-only memory and can thus never be modified.
Boot1 can be written, but various read-only checks prevent anyone from doing so.
Boot2 however, can be modified on older consoles due to a bug in boot1 on those systems. Since boot1 cannot be modified, this means that any "BootMii as boot2" compatible Wiis will always be compatible. Unfortunately, Nintendo later patched boot1 on newer consoles, making those consoles always incompatible.
If your Wii is compatible, BootMii will replace the normal boot2 code. This allows it to execute some custom code before the Wii menu boots, meaning it can serve as brick protection. Of course, the boot2 code is limited in size, so instead it loads certain files on the SD card instead. These files are generated during the "prepare SD card" steps.