Tivo is a DVR recorder, it's just a brand name one. It works best with broadcast tv, and some cable companies. Most cable and satellite providers have their own DVR machines, which can vary widely in how their controls and such work. Depending on company some want you to buy the box, others you just pay for the services ie rent.
It works kind of like a VCR but you can rewind while it is recording without messing up the recording. Some companies also let you record two channels at once so no more missing one show due to time slot issues. Most of them store your recordings for as long as you set them to, and most can be set to automatically record every episode of whatever shows you want.
My biggest issue with mine is I can't set the quality, so sometimes the recorded picture goes pixelated because it wasn't recorded at the best quality, ie oceans and other smooth colors have a tendency for this.
--Mia