The one thing that binds all GNU/Linux distributions is the kernel (whence the Linux name). Beyond that, the X window system provides communication with the kernel for the desktop environments (Kde, Gnome, Xfce and others). The set of applications the distribution (distro) provides to make use of the kernel is what makes up the distro.
The packaging of applications falls into these categories: source code (packages with .tar, .tar.gz, and .tar.bz2 file extensions), .tgz for Slackware and derived distros, .deb packages for Debian and Debian-derived distros (such as Kanotix on the eeePC), and .rpm for RedHat and other .rpm-based distros.
All distros may use source code apps (which must be compiled in order to install). Your eeePC probably came with a stripped-down version of Kanotix, which is Debian-derived and uses packages with the .deb extension.
The usual way to install software is to first update your Debian repositories list to get information on the available packages. Open a console and enter the command 'apt-get update' to update your repository information, then 'apt-get install <packagename>' to install a package. If there are dependencies (the package requires that other packages be installed in order for the one you want to work) apt will also install the dependencies, assuming they are in the repository.
You can also do the same graphically with the Synaptic program, which is a graphic front-end for apt-get. It will list the available packages in the repository, and indicate which you now have installed.
To remove a .deb package, issue the command 'apt-get remove <packagename>', or use Synaptic to select the application to remove.
Rpm based distros have similar means of installing/removing software.
Source code based distros also have package management software.
The major differences between windows and Linux: you seldom, if ever need to defrag (and then it's usually not necessary until a partition nears full), and stability (you can leave the system running for ages). The main difference, in my opinion, is choice. You have choices you could never dream of in windows. The second great difference is the cost (all you need is an internet connection, a cd/dvd drive, and disks to burn the software onto, and you can have gigabytes of software).
The drawbacks: hardware vendors don't support Linux as readily as they do windows and Mac, so getting drivers for bleeding-edge hardware can be a pain. Also, you don't have an undelete utility such as in windows. So, if you delete something, getting it back is very difficult unless you have a backup you can use to restore what you deleted.
