FreeNX is what you're looking for. It is to Linux what Remote Desktop is to Windows. FreeNX is, as the name indicates, is free. I've never used RealVNC, but it also appears to be free and is available for Linux. You may want to try TightVNC, as it's free.
VNC and FreeNX don't do quite the same thing, so you should pick the one that does what you're looking for.
VNC will allow you to access the same desktop that you would be using if you were sitting at the computer. If you were to open Mozilla, for example, while sitting at your home computer and then go to a different computer and VNC into your home computer, you would be able to see and use that instance of Mozilla that you had opened.
FreeNX, on the other hand, gives you a virtual desktop on your home computer when you are at a remote machine. You wouldn't see that instance of Mozilla that you had opened at home. Anything you do while logged in via FreeNX is contained in that session.
In my experience, FreeNX is much more responsive than VNC. It feels more like you're sitting at your home computer.