I have been reminded that many readers coming here for the first time won't have a clue what Linux is, much less the difference between the many Linux 'distros' (releases), or the many Linux desktop environments.
Much as you think Windows looks different from the Apple OS-X desktop, wait til you see the variety of Linux desktops...
The advantage is that with a little knowledge, you can choose the desktop environment that suits you and your hardwae best; install it on a 'try-before-you-don't-buy' basis (they are all free software) and easily uninstall the desktop you don't like.
As has been said elsewhere, the choice of Linux desktop environment shouldn’t affect the stability and structure of your operating system. That is dependent on the underlying Linux distribution you are using.
Desktop #1: GNOME
Still the most popular desktop environment in the solid and workman-like Gnome 2.x release, the latest Gnome 3.0 with the radical Gnome Shell has dented credibility with the Gnome faithful.
In a standard two-panel configuration, Gnome 2.x gives menus and multiple workspaces; Gnome Shell is a bit more quirky in operation as well as demanding a chunky graphics card. There's a decent but uninspiring file manager in Nautilus. There's also a vast number of Gnome-compatibile applications written using the GTK2 and GTK3 development frameworks.
KDE is the second most popular Linux desktop environment, despite the hiccup when KDE 4.0 was released in a disastrously buggy and unstable release. KDE 4.6 is back to it's best, featuring stunning graphics, anmated effects, 'Plasmoid' widgets as desktop addon's (Windows Metro live-tiles and then some). The Dolphin file manager is good and Rekonq, the KDE web browser, stunningly under-appreciated.
Thanks to it's large user and developer base, KDE is well-stocked with programs developed on the KDE framework; see a program name beginning with a capital K (and there are some bizarre 'K'-names), that'll be a KDE application. It's possible but problematic to mix GNOME and KDE application stacks.
Low-powered machines may struggle to run KDE, especially with the many desktop effects enabled.
Desktop #3: XFCE
XFCE is the “lightweight” alternative, as in it uses a slimmer software stack than the previous two. Lighter, smaller desktop stacks are available. However, XFCE scores highly for several reasons. It is a fully functioning desktop environment with low-memory usage, making it fast and responsive. It is easy on the eye without the full range of fancy effects and animations, making it a good substitute on lesser hardware.
You can set up a conventional two-panel-plus-menus desktop with a lot of flexibility. XFCE uses the same application framework as GNOME, so the full collection of GNOME applications will run seamlessly and it has a growing library of lightweight XFCE alternative programs. The Thunar file manager does all the basics very well. A lot of defectors from KDE 4.0, Gnome 3.0 and Unity (below) have switched to XFCE.
Desktop #4: Unity
Unity is the departure for the Ubuntu Linux distribution - currently the number one by number of users, Ubuntu's lead has dwindled lately, with unkind critics blaming the Unity interface. Based on GNOME foundations, the desktop itself has gone very Mac-like, with a dock on the left side and a custom dash.
Unity is as radical a departure from conventional GNOME 2.x as GNOME Shell; the developers Canonical are making a bold statement about usability and design, although this change and the increasingly corporate, locked-down design process has alienated some of Ubuntu faithful. If you haven't used computers much before, you shouldn't have a problem getting into Unity; switching from something else... it might slow you down.
There are still more Linux desktop environments you can try - LXDE, OpenBox, Fluxbox, Enlightenment and others - more or less like the big four, lighter, heavier or subtly different. RC