Gnome, KDE and XFCE are the three most popular "desktops" under Linux. In Windows there's only one desktop (explorer) and you can't really change it, so it's hard to explain...
Basically the operating system is made of different layers...
- the kernel (deals with memory and hardware access)
- the core (I'm vulgarizing a bit here.. everything which makes the OS run under the hood)
- the X server (manages the mouse, and the screen display)
- the window manager (manages the location of windows and the focus)
- the desktop (manages the look and feel of the windows and the panel, and extra tools that go with the desktop... the file browser for instance.).
Basically Linux Mint is using Gnome by default. But we also have two community editions, one coming with KDE and one coming with XFCE... and because some applications are better suited under KDE or under Gnome, these editions come with different software selections.