Jerry - you use them for remote controlling a workstation or server. You take over the screen of the other machine. Great for managing servers from your office.
Julian - I have found the 1.3dev6 version to be unstable on my server, it has even made the OS reboot. I have moved back to the last stable version which seems ok.
Jens is right on about what it does, and VNC/TightVNC is free and open-source and cross-platform to boot. There's even a Java applet you can use as a viewer on a web page, so you don't even have to have a viewer client installed on your machine. You just access the server via a web browser from a certain port, login, and away you go.
The big downside to VNC has always been that it's pretty slow compared to the commercial options like PCAnywhere, NetOP, etc. The TightVNC offering (which is a branch of VNC) is very competitive in terms of speed, and it includes some nice options like file transfer.
If you're wondering why you'd use TightVNC instead of Remote Desktop on a Windows server, just think of this: what happened the last time you tried to open a Notes client on a Windows server via Remote Desktop? Also, could you see the Notes server command window? Could you share a connection with someone else to show them how to do something?
Anyway, Jens: I hope they fix the problem you're having by the time 1.3 goes gold. The dev6 version hasn't given me any problems yet, but I've only got it running on a basic test server and a Windows XP box, so I haven't put it in any unusual situations. I wonder if it had something to do with the new mirror driver?
Please make sure that you're using the latest version of DFMirage driver. The most stable and WHQL certified version of video hook driver for TightVNC can be found on DFMirage webpage.
What is VNC or TightVNC good for again?