Remote control of a Mac that's behind a NAT I'll soon be traveling to set up a Mac for my elderly mother. I'll need to later be able to control it from outside the network. I already use Apple Remote Desktop to control other Macs on my network, but i don't know how to penetrate the remote network to get to her computer.

What do I have to do to get her router to let me connect to it? I don't know what type of router she has, but if need be I will be installing an Airport Express or Extreme instead.
Without VPN-Functionality on the Router you have to rely on Software like 'LogMeIn' or 'Teamviewer' to access you Machine from outside your Home-Network.

As an alternative to LogmeIn you can use "Teamviewer". You have to install 'Teamviwer Host' on the Machines you want to control from outside and the regular 'Teamviwer' on your Machine.