MythTV is an open source DVR / PVR (what's the difference?) that runs on Linux. Definitely not a project for a first time linux user, but there are different packages available that make installing it very easy. The hard part is getting the right hardware together to get it to work right. Once you do though, the setup is pretty sweet. Nothing like owning your own solution instead of renting it from your cable provider.
A couple of great distributions that are built solely with the intention of making it easy to get mythTV set up are MythDora and also MythBuntu.