working on a remote Mac How can I log on remotely from my Macbook Pro to a friend's iMac so as to share her screen, and help her fix things when she has problems? My own knowledge is not very profound, so please bear that in mind with your answers. I am somewhat more at home with Unix than with Mac Os, but still no expert.

I realize that I would at least need her Mac's IP address. However, I think her computer gets its IP address from DHCP, so it would presumably change every time she turns her computer on or off. I suppose she could email me the IP address when she has problems, but that's not ideal.

Do I need to buy software for this task, or can I manage with facilities provided as standard by Apple? Does it matter which OS we work on? I am still working on 10.6.8, and I'm not sure what her iMac is running.
There are many ways to achieve your goal. Apple has its own product: Apple Remote Desktop.

But in your case I would suggest another tool called Teamviewer. It is free for non-commercial use and easy to handle. Both sides (you and your friend) have to install the teamviewer application. When startet, you don't have to bother with IP adresses or whatsoever. Teamviewer will give you an ID and a password which is enough to connect to your friends mac.
It is platform indipendend and will work also on lots of Linux distributions.

Another nice, maybe even easier to use but non-free service is Gotoassist. This service does not need to install software on the remote machine. The remote machine just has to open a specific link in browser and you're done.

As I mentioned before, there are many other possibilities depending on the type of support one wants to offer, for example: SSH or VNC.