BIOS is an acronym of Basic Input/Output System. It's the program that tells DOS (and Windows 95, but to a lesser extent) how to communicate with the various peices of hardware on a computer. It also contains routines that find out where the boot programs are and load the operating system from a disk.
There are two parts of the BIOS that people usually see. One is the Power On Self-Test (called POST for short) which runs every time the computer is started. The POST sequence goes through its list of what hardware it thinks the computer should have and makes sure everything is working properly.
The other part is the Setup Utility, also referred to as the CMOS Setup or just CMOS for short. CMOS is the engineering name for the type of memory which holds a list of hardware and configuration settings. This memory is kept turned on 24 hours a day by a little rechargeable battery inside the computer, because if it ever lost its settings, the BIOS wouldn't know how to start the computer.
A big improvement in BIOS chips was the move from ROMs to Flash EEPROMs (which means Electrically Erasable Programmable Read-Only Memory). With standard ROM chips, the only way you could upgrade your computer's BIOS would be to open up the computer, remove the chip that's plugged onto the motherboard, and put a new chip on, hoping that you don't break any pins or electrocute the computer in the process. With a Flash BIOS, upgrading the chip is a simple as upgrading your software. You simply put in a floppy disk and run an installation program.
A word of advice: if your system allows it, make a backup copy of your current BIOS to an emergency recovery disk. That way, if something goes wrong with upgrading the BIOS, you can still get the original one back.
Newer computers have BIOS's that are Plug and Play compatible, meaning that if you have hardware in the computer which is also Plug and Play compatible, the POST sequence will find that hardware and fetch the settings itself without you having to tell the Setup Utility what it is. A very common example is hard drives that are autoconfigured. When you add such a hard drive to the system, the POST will ask the hard drive how many cylinders, heads, and sectors it has, whether to use LBA or ECHS translation, and a number of other things that on older computers you would have to manually type into the Setup Utility.