PersonalVideoRecorders, or PVR's, are computers which are connected to your TV which will allow you to record and watch shows broadcast on TV. Commercial products like Tivo and ReplayTV were the first instances of this that I'm aware of. The problem with the commercial option is that I now live in NewZealand and none of them have access to the programming guides here. Being able to tell your PVR to record "every episode of 'The Simpsons'" is the main feature I want from a PVR so this makes them pretty much worthless to me.
See also: MythTv, Sam Ruby, DigitalMediaManagers, UniversalPersonalProxy, PersonalDataRepository, DigitalPalace, WhenThingsStartToThink, MyLifeBits, AsWeMayThink
Open Source Solutions
Recently there have been some OpenSource attempts at building PVR's which look fairly hopeful:
http://geexbox.org/ (only a movie player, not a recorder)
There is already an attempt at getting programming information into an open format called XMLTV, and they claim to support NewZealand programming (w00t!):
The next piece will be finding suitable, and affordable hardware, for this device. The main unusual requirement I have is that it be quiet as it will live in the living room. If I can find something powerful enough perhaps it could also end up being an MP3 juke box and MameCabinet.
More as I investigate.
More generic information on hardware and software required for a "Home Theatre PC" system, which is a super-set of what I was originally talking about.
Random Documentation and Howtos
RobFlickenger's description of how to setup a diskless box to do video playback onto your TV using GeexBox. http://nocat.net/~rob/hackery/slimremote.html
Tivo in New Zealand
It's recently become possible to use a Tivo in NewZealand. It's still a bit rough around the edges but may be worth investigating ...