Run level 0 is the system halt condition. Nearly all modern X86 computers will power off automatically when run level 0 is reached. Older X86 computers, and various different architectures will remain powered on and display a message referring to the halt condition.
Run Level 1 is known as ‘single user’ mode. A more apt description would be ‘rescue’, or ‘trouble-shooting’ mode. In run level 1, no daemons (services) are started. Hopefully single user mode will allow you to fix whatever made the transition to rescue mode necessary.
(You can boot into single user mode typically by using your boot loader, lilo or grub, to add the word ‘single’ to the end of the kernel command line).
Run levels 2 through 5 are full multi-user mode and are the same in a default UserLinux (Debian) system. It is a common practise in other Linux distributions to use run level 3 for a text console login and run level 5 for a graphical login.
Run level 6 is used to signal system reboot. This is just like run level 0 except a reboot is issued at the end of the sequence instead of a power off.
In the interests of completeness, there is also a runlevel ‘S’ that the system uses on it’s way to another runlevel. Read the man page for the init command (“man init”) for more information, but you can safely skip this for all practical purposes.