Firmware is software that's stored in non-volatile memory, most usually some sort of flashable EEPROM. This means that when the system's turned off, it doesn't vanish like sofftware stored in regularly memory. I suppose ROM memory can be considered firmware (*very* firmware), but nowadays it's more common to see firmware stored in memory that can be updated by special software to fix bugs, improvements, and so on.
As others have said, it's generally associated with the very low-level system software that gets a Mac or PC's hardware initialized before the operating system is booted. It's also pretty common in anything electronic that's complex enough to require a proper microprocessor or microcontroller.
Looking around the room - my old Axis 2100 network camera has firmware (a complete Linux implementation, no less), my router has firmware, my Axis Officebasic printserver has firmware, but might as well not have because they've never issued an update for the damn thing. Your cellphone may have firmware (my Nokia does) although getting upgrades out of providers is like getting blood out of a stone.
Anyone know if the iPod actually has true flashable firmware? Or does it take the evil approach favored by the Nomad Jukebox. This has very basic firmware that boots the actual software that the unit runs off the internal hard drive, thereby making upgrading an extreme pain (since you have to clone the appropriate sectors of the hard drive, using something like 'dd' on a Linux box, and copy the resulting binary to the new drive)