Quote: "Is this a plan by the Brits to take over America?"
Nah, we'll just buy you out after you default

. Sorry, that was mean.
Anyhow I'll try and give you a quick run down:
The things you mension are all programming languages. That means that they are used to create the logic of your games. They do not provide integrated engines for making graphics and sounds etc. However they do provide commands to load and display/play graphics/sounds.
DarkBASIC Pro
Advantages: Easy to learn, powerful commands based on DirectX 9, huge range of commands ranging from 2D/3D graphics through to physics and networking, good plugin system and community support so anything missing from the language can also be added. Very flexible and so almost any type of game should be possible.
Disadvantages: Relatively slow (although more than adiquate for most things), simplicity does mean complex but powerful features such as Object Orientated Programming are missing. Programs will only run on Windows.
DarkGDK
Advantages: Written in C++ and so normally runs faster than DBP programs. Allows you to use DBP's powerful and simple commands from C++. C++ means complex features such as OOP are available. As it's C++, you can benifit from any C++ libraries, not just those made in to plugins for DBP.
Disadvantages: Relatively difficult to learn. DBP's commands arn't Object Orientated or Thread Safe themselves and so you can't use all of the power of C++ with them. Programs will only run on Windows.
PureGDK
Advantages: Use DBP's powerful commands. Can be used from a growing range of different languages meaning you can chose one you like and that suites your needs. Opens some exciting possibilities such as using DBP commands in a Java Applet in your web browser. Depending on the language you use PureGDK from, you will probably gain the more powerful structures and faster code of DarkGDK although not necessarily.
Disadvantages: Largely the same as DarkGDK. Reliance on DirectX 9 will still confine it to Windows.
AGK Tier 1
Advantages: Write your code once and deploy it to loads of different platforms (iOS, Windows, Mac OS X, Meego, Bada) and more in the future (Android?, Web Browsers?). Easy to learn. Powerful 2D and networking commands. You get tier 2 included when you buy it.
Disadvantages: No support yet for 3D. Not yet released (although it will be in under a week). Lack of powerful and complex structures in the same way as DBP. Relatively slow although early reports suggest it runs fine. No plugin system and so you are reliant of TGC adding any commands that are missing that you need.
AGK Tier 2
Advantages: Use of AGK's powerful commands in C++. More complex structures provided by C++. Will probably run quicker. The same commands can be used in writing for all the devices supported by AppGameKit Tier 1. Anything missing from AppGameKit can be implemented in C++.
Disadvantages: Relatively difficult to learn. May need to tweak your C++ for different platforms although the AppGameKit commands should not need to be changes.
OK Well I hope that helps a bit but I should point out that:
1. I'm not the world expert of this so you may get a better answer if you wait a bit
2. A lot of that is a matter of opinion
3. I've only actually used DBP and DarkGDK so what I say about PureGDK and AppGameKit is from what I've read about them.
Anyhow if that doesn't make sense, do post any follow ups and I or someone else will have a go at answering.