BIOS stands for Basic Input/Output System.
This is software that usually resides on a PROM chip on your motherboard (making it an example of firmware). It is the software (or OS if you will) for your hardware and peripherals and allows your main OS to talk to your hardware. The exception is block devices like disks, which most OSes will talk directly to for performance reasons (simpler OS like DOS still uses BIOS routines).
The BIOS configuration process allows you to set up many of your peripherals, as well as your hard drive geometry and system clock. The configuration file(s) for BIOS are stored on the CMOS non-volatile RAM.
The BIOS can be updated in a process known as flashing. This term came about because earlier versions of the BIOS PROM had to be blanked out by a UV lamp before being overwritten. Flashing a BIOS is one of the riskiest processes that can be done to a computer (right up there with replacing the processor) because a mistake will render the motherboard unbootable. Flashing a BIOS has gotten a lot easier over the years, but be sure to plug your computer into an UPS to make sure you won't have a power outage during the job.
The BIOS is important to the process of securing a system, because it controls what order drives are checked for booting. If a malicious user has physical access to the system, and is allowed to change settings in the BIOS, they can boot from a CD-Based Distribution of Linux (or any other operating system running on that machine) and defeat any security you may have on the system. Even with encryption, they can transfer an image of the drive to other media or over the network for cracking at their leisure. Thus, it is important to set the BIOS password that prevents changing the settings. But note that given physical access to the machine, the aforementioned malicious user can reset the BIOS using the given jumper settings and erase your BIOS password. Because of this, it is important to practice good physical security on sensitive machines.
The BIOS was originally a proprietary system developed by IBM. If it hadn't been reverse engineered by Compaq, the world of personal computing would be very different today. Until cheap clones were made possible by the reverse-engineered BIOS, IBM took advantage of its monopoly position to keep prices high. Perhaps you would be using an Amiga right now instead?
The LinuxBIOS Project is attempting to use the Linux kernel as an Open Source alternative BIOS. The effort is still highly experimental, so don't try it unless you have a motherboard you wouldn't mind using as a doorstop.