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Does anyone know how to import footage from games? whether it be replays, cut scens etc?

Do you mean computer games (such as Flight Simulator) if so you need to record the game as you play it with some third party software such as
FRAPS
or Camtasia Studio

yeah. Well from Xbox console.

What does this software do exactly? how does it work?

Sorry but I dont have an Xbox so cannot comment upon how you would attach one to a computer never mind record it.
The two links I gave (There are many other similar programs but those two are the more famous) will record whatever is on your computer screen.
In the case of Camtasia Studio it will also record the mouse.
The primary function of Camtasia Studio is for creating presentations or tutorials.

I might want to show you how to do something with VideoStudio and so instead of simply writing a thesis with or without screen shots, I could record the screen as I am doing it, I can even connect a microphone and give you an audio commentary at the same time.

FRAPS is more purpose built for recording games being played on a computer.

Both record the screen, Camtasia Studio is orientated towards presentations, FRAPS is orientated towards recording games.

Probably the simplest way would be to use a camcorder and a tripod. That is if you already own a camcorder.

Another way would be to purchase a video capture device. Dazzle seems to be the most popular one out there. You can pick one up from your local Best Buy, Circuit City or Staples with prices starting as low as $49. It connects to your to PC through USB and it has all the composite hook ups you'll need.
Here is the tricky part -
You'll need to split the signal coming from your game console unless you're using a high performance PC. This is because if you try to view your game play on your PC monitor while recording you'll run into problems with LAG. Depending on the CPU power of your PC there can be as much as 1.5-2 seconds of lag [example: you press the "A" button but you won't see a response until a second later]. This lag comes from the video capture device's encoding process. So ideally you'll need to purchase 3 RCA/composite splitters from your local Radio Shack. This way you'll be able to play your game without the lag [you'll have the signal going to your PC and your TV at the same time.]