USB - Universal Serial Bus. This transmission standard goes back to the mid 90's, where Windows 95 (with USB support) was first introduced. Back then, about the only thing you could find with USB were joysticks. Soon though, the speed and efficiency was realized and just about everything became USB connected.
USB 2.0 was faster yet.
USB 3.0 is now the standard and screams.
Firewire (Firewire 400, as it's called) was the standard for tape based camcorders to communicate with each other, or to a computer or hard drive to store the data. It's faster than regular USB. The 6 pin connector on the computer side is notorious for destroying many a camcorder. If the cable is inserted at an angle (or even backwards - yes, it does go in backwards with just a little help) then the 12V DC that is supplied at the computer side travels down the wrong pair and instantaneously fries the firewire IC and associated components. There is no smell, no smoke, no sound. Just stops working. Makes for expensive repairs!
Many times we'll repair the camcorder and send it back to the customer, only to have them do the exact same thing again! This has happened countless times, and the manufacturers are very well aware of this. Once a cable is inserted backwards, it's easier to do it again. The best thing to do is to actually look at the back of computer and while watching, plug in the 6 pin end of the firewire cable. Leave it plugged in, and just connect and disconnect the camera side whenever you want.
There is no need to power off the camcorder or the PC to connect the firewire cable, despite what you've been told. I've been at this a long time - I should know!
Firewire 800 - A newer and faster version of firewire that uses a square connector, versus the problematic D connector of old.
Firewire 1600 - Faster version still and competes with USB 3.0. Backwards compatible with Firewire 800. Mac uses this version
Firewire 3200 - While this version does exist, it really didn't get into any production. It's also suppose to be backwards compatible with Firewire 800 and 1600.
HDMI is now used for HD camcorders, HDTV's and BluRay DVD players so that they all play nice together. It's simple really. One cable that carries both audio and video - including Dolby stereo. Now what does make a difference is the cables. Almost all are made in China, but there are the dollar store varieties and gold plated, diamond encrusted versions. No need to buy the later - but spending more than dirt on a cable is a good idea. If you're watching a movie and the sound keeps cutting out, or there's intermittent problems with the video - try a higher quality cable!