The first DVR system, TiVo, was introduced in 1999 (along with a similar service called Replay that wasn't quite as successful). In this model, you pay for cable and any premium channels you want, which run through the TiVo box before reaching your TV.
This allows you to choose your programming from a very sophisticated menu system, record your favorite shows, receive recommendations based on the things you like and set a variety of recording options. You can search by actors or even subject matter, then set the program to record just once, every time there's a new episode or every time the show comes on.
A few years ago, AT&T released its own version of a similar service which was actually integrated into its cable-provider network, a system called U-Verse. That means you don't have to buy a separate device or subscription in order to get the same programming and recording power.
In both cases, the advantages are clear: The companies have spent so much time on their recording, recommendation and programming software that with a touch of a button, you can record reruns of your favorite shows on every channel, set rules on how many episodes to save, let the box determine what shows you might be interested in trying, and trust that the software will remember and learn about your preferences over time. It's a great way to learn about new shows while making sure that you never run out of recording space.