A firewire port is a high-speed data transfer port - it's a standard also known as IEEE1394, SB1394 and iLink among other names. Firewire can be used for very high speed transfers such as external hard drives, CD-burners, video cameras.
Merlin is correct that firewire often comes on add-in cards, but it can also be built into your motherboard (including laptop's motherboards). There are two type of firewire port - a larger six-pin port and a smaller four-pin port. Both contain four-data pins but the six pin ports also run power through the cable - this can power low-demand external devices such as micro hard disk drives, without the bother of having an external power brick for the drive.
The most common use of firewire is (as Merlin has suggested) digital video - very often digital video cameras come with firewire built in.
USB2 is similar to firewire in many ways (similar high speed data transfer), but is superior in some ways (backward compatible to the almost univerally implemented USB 1.1 standard) but inferior in others (USB2 transfers in 'bursts' that mean whilst it is theoretically faster than firewire, in actuality it tends to be a bit slower).
Firewire also tends to provide more solid power and can power devices that a USB port cannot - but this is to some extent also dependent on the device you plug in.
