The X Window system is basically like the GUI of Windows. It keeps you from having to stare at a command prompt all day. It also provides numerous services to applications, such as hardware abstracted graphics acceleration and the like.
As for why your desktop is different, you were logged in as root when you were at that command prompt. When you see a pound sign (#) as the prompt, you are root. When you see a dollar sign, you are an unprivileged user (in this case, kuroneko).
Don't drop out of root yet; drop back into a normal command prompt and type "nano /etc/X11/xorg.conf".
I don't remember what Debian's default configuration looks like, but they're generally all pretty similar. In the Module section of the file, make sure that these two lines exist:
Load "glx"
Load "dri"
If they exist but are commented (i.e. have a pound sign in front of them, just like MegaZeux's config), uncomment them.
Now stroll down to the ServerLayout section. Take a look at what the Screen line says. Now, look for the Screen section that has that identifier. When you get there, look at the Device line. Now, as you may have guessed, you're going to want to look for the Device section that has that name.
Somewhere in that Device section you should see a Driver line. Make sure that the driver is set to "i810".
Now, at the end of the file, add the following lines if they aren't already there:
Section "DRI"
Mode 0666
EndSection
That section allows non-root users to use hardware acceleration.
Alright, now try to start X.
If X fails to start you'll probably need to install the i810 module. Start up aptitude, type forward-slash, then "i810" to search for it. Repeat the search (just press forward-slash, then enter, and it'll give you the next result) until you get to the appropriate package (it's called "xf86-video-i810" in Gentoo; I think it has a similar name in Debian). Press shift+=, then 'g' twice to install it. Quit aptitude and try starting X again.
When you get X running, do the "glxinfo | grep direct" again.