An important run level is single user mode (run level 1), in which only the system administrator is using the machine and as few system services, including logins, as possible are running. Single user mode is necessary for a few administrative tasks, [1] such as running fsck on a /usr partition, since this requires that the partition be unmounted, and that can't happen, unless just about all system services are killed.
A running system can be taken to single user mode by using telinit to request run level 1. At bootup, it can be entered by giving the word single or emergency on the kernel command line: the kernel gives the command line to init as well, and init understands from that word that it shouldn't use the default run level. (The kernel command line is entered in a way that depends on how you boot the system.)
Booting into single user mode is sometimes necessary so that one can run fsck by hand, before anything mounts or otherwise touches a broken /usr partition (any activity on a broken filesystem is likely to break it more, so fsck should be run as soon as possible).
The bootup scripts init runs will automatically enter single user mode, if the automatic fsck at bootup fails. This is an attempt to prevent the system from using a filesystem that is so broken that fsck can't fix it automatically. Such breakage is relatively rare, and usually involves a broken hard disk or an experimental kernel release, but it's good to be prepared.
As a security measure, a properly configured system will ask for the root password before starting the shell in single user mode. Otherwise, it would be simple to just enter a suitable line to LILO to get in as root. (This will break if /etc/passwd has been broken by filesystem problems, of course, and in that case you'd better have a boot floppy handy.)
| [1] | It probably shouldn't be used for playing nethack. |