Is there an equivalent to Remote Desktop Connection that I can use to remote into my Ubuntu desktop at home from a Windows machine? Preferably this would be free.
You can try NX-server and client. They have free linux server, and the client is free as well.
3 things I like more in NX than using a normal VNC solution:
Thinks I do not like: nxserver is not opensource.
Compromise: use freenx server on linux (opensource), and nxclient for windows (free as in beer).
If you are looking to launch programs from your Windows machine you will need:
Xming (Windows X Server) - http://www.straightrunning.com/XmingNotes/
And then just SSH to your computer, but in Putty be sure to check the box "forward X11 session". After you have a terminal you should be able to launch programs like: