When a computer starts, the number of beeps the BIOS outputs tells you the state of the computer. On some computers, one beep means all is ok, but 2 beeps mean there is an error. LILO uses the same kind of codes.
The number of letters you see from the word LILO on the screen says what is wrong. The whole word means everything is fine, only LI means only the first part of LILO could be loaded. A full description of this is available from the Bootdisk HOWTO.
When LILO can't load, it's a major problem. This often means that the boot code was corrupted. The only way to boot is from a floppy disk. In RedHat, you can use the rescue disk, in Slackware, you can use the boot disk with the "mount" image.
When LILO is fine, it's often easier to figure a boot problem. If the kernel panics when it tries to boot, it is usualy due to a configuration error. You can tell LILO to mount another kernel you may have, like a "safe" or "old" image you kept for these cases. If the problem is in initialization scripts, you can tell LILO to boot directly into a shell with the following boot command line:
LILO boot: linux init=/bin/sh
Where "linux" would be your kernel image.