Over the Air
Network TV broadcasts programming for free. It used to be that cable might provide you a better picture, but with digital TV you either get a perfect picture, exactly as good as cable, or you probably get nothing at all. If your area is well served by TV stations, this is a workable solution.
If you have been using cable but want to save money, digital over the air TV uses the same antennas (because it uses the same frequencies) as old analog TV. Since the signal does not degrade gracefully, you probably need an antenna on your roof. You will get the same signal as your neighbors, so you can ask around and find out what to expect.
Cable
A cable TV system is immune to interference. It transmits digital data without the error correction and redundancy used by over the air, so it can get twice as much data or 40 megabits per second on each of its 158 TV channels. This allows cable to carry two HD programs and four SD programs on each of the 158 frequencies, or up to 10 SD programs on a frequency with no HD.
All the network broadcast channels that you might or might not be able to clearly receive over the air are also available over the cable with excellent reception. These free to air programs are not encrypted and can be received with any device that supports Clear QAM content. However, the FCC has approved a request to allow cable companies to encrypt all their content in the future, so this is a bad time to buy new cable devices that do not support CableCard.
The digital TV channels that you pay a montly surcharge to receive (USA, SiFy, BBCA, TNT, MTV) and all the premium movie channels (HBO, Showtime, Starz) are encrypted and require either an external set top box or a CableCard tuner.
Verison FIOS also supports CableCard even though it is not technically a cable service.
Satellite
Satellite TV systems (DirecTV, Dish Network) and ATT U-Verse do not support CableCard tuners or Clear QAM. The signal has to be processed by an external proprietary set top box provided by the company. Therefore, you can only record the output from this external box, and currently only the Hauppauge HD PVR external box or the Hauppauge Colossus card adapter can record an HD picture generated by an external tuner box.