Yes you can record TV shows, movies and whatever that is coming from your cable TV. With these devices it takes a standard coax cable input.
There are a few options depending on if you want a an internal or external device. All will require renting a cable card which decrypts the cable signal.
For a basic external TV tuner box with 2 TV tuners.
http://www.hauppauge.com/site/products/data_dcr2650.html
3 tuners external:
http://www.silicondust.com/products/hdhomerun/prime/
That silicondust product looks the most suitable for me. Where does it record TV? on you computer or does it have a big hard drive?
Internal card/6 tuners:
http://cetoncorp.com/products/infinitv/
No remotes on these models but the 2nd and 3rd options have smart phone apps.
Be seeing you, the Prisoner...