Back in the day, people had no options for recording TV shows. Each show broadcast whenever it was scheduled, and if you wanted to see it, you had to be firmly in front of your TV set at the appointed time. The VCR revolution changed all that, and the advent of digital technology took the concept of recording to an entirely new level. Nowadays people have a number of options for recording their favorite shows, including some "obsolete" ones that still work just fine.
Digital video recorders like TiVo remain the go-to technology for recording and storing television programs. They essentially act like a computer drive, saving the TV show as a digital file and storing it for future playback. They also contain a number of other advantages, including the ability to record an entire season's worth of shows with just a single touch of a button, and smart features that find and record shows you might like based on your overall viewing habits. DVRs tend to contain much more storage space than other recording mediums as well. Their only drawback entails their non-portability: once you fill up the hard drive, you either have to record the shows on another medium or delete existing shows to make room.
DVD recorders allow you to record TV shows on a DVD and play them back later. They don't have as much storage space as a DVR--each DVD only holds a few hours' worth of programming--but because each DVD is self-contained, you don't need to worry about running out of space. Recordable DVDs come in two distinct types. DVD-Rs may only be used once; the show is "burned" onto them permanently. DVD-RWs, on the other hand, may be used to record and re-record as often as you like: erasing earlier shows the way you would with a VHS tape. DVD-Rs tend to be cheaper than DVD-RWs, which constitutes their most tangible advantage.
Though it has almost gone the way of the dodo, don't count out the venerable VCR when considering recording options. VHS tapes suffer from a number of disadvantages: limited recording space, an inability to record shows in high definition and a delicate medium that degrades much more quickly than other formats. On the other hand, they are extremely inexpensive these days, and people who may be intimidated by newer technology still know how to program their VCR fairly easily. VHS tapes also share the benefits of DVDs in that they don't have any hard drive space to fill up. As long as you have more tapes, you can record as many shows as you want.