Can I manage another computer in over an Internet connection?
Apple has tools for that in Sharing.

I personally am a fan of the app called TeamViewer and use it with my family and extend family and friends in Europe as well.

It is no so much Sharing but taking the control and fixing stuff remotly, or just watching them do something as a teacher.

It can be set to auto start with mac so you can access any time without need for the other party do to anything.

It of curse offers Audio/Video chat or messaging while in session, but I mostly use that to tell the other party to stop moving the mouse around and just watch me fix they problem :)