There are several types of cards that support several types of video stream inputs. First we'll go over the type of signals, then the cards, then which cards support which types of signals.
A regular TV tuner card is one that accepts NTSC video and sends data to the CPU to encode into digital video. It's a pretty basic tuner card and very cheap, around $20-$40. Cards will have one RF TV antenna coaxal port, S-Video port, or Composite port, two of these types, or all three. These cards will require you to run a jumper cable out of the tuner card and into your sound cards line-in port for sound though. These cards usually only can receive NTSC TV over the air (OTA) or analog cable, or video by way of the S-Video or Composite inputs. You can connect a satellite, cable, or game console set top box to either one of these inputs to capture video from the device.
For a fairly complete list of NTSC, PAL, and other supported cards under Linux and more information about them than you'll need see Gunther Mayer's card listing page. I recommend checking major computer hardware suppliers online (CompUSA, Circuit City, Walmart, etc) and find out which cards they have. Then visit the supported card pages above and see if the card is supported. You will want to be aware of which inputs the card has and if it has stereo or mono sound (Many cards just have mono sound).
Another very good database of cards and reports from the users ares at pvrhw.goldfish.org. Not only is there information about the cards, there are also reports on users complete pvr systems and some quick install guides for specific hardware. This is a very good read too.
Hardware encoding TV cards will do the same as a regular TV card but also encode the video and audio on the tuner card before it sends it to the computers system bus. This offloads all encoding from the CPU so the CPU can be used for other things. This is helpful on systems with slower CPU's or if you don't have a sound card in a system. These type of cards are very popular, of which the Hauppague brand is the most popular. The Hauppauge PVR-150 is a low cost tuner, the PVR-250 is a step up in quality, the PVR-350 has a S-Video OUT port which will do hardware decoding/playback of video recorded by the card to be displayed on the TV, and the PVR-500 is a dual tuner card. Buying the card online is usually the cheapest way to get one of these cards.
Since cable signals can be encrypted, some cable companies sell cable card tuners that can be inserted directly into a TV and used to decode and display cable TV, this is rare now days. There were illegal tuner card readers available for the PC years ago but I'm not aware of them being sold any more. However, by law, all local channels must not be encrypted and most cable companies only encrypt premium channels, making it possible to view most cable channels. There are three common types of Cable, Analog, QAM64 and QAM256. QAM 256 is the most common nowdays and is only supported by some cards.
To start, HDTV is three times better quality than DVD and eight times better quality than TV, and that is if you were watching a perfect NTSC signal which isn't possible. HDTV uses the same type of digital format used on DVD's to guarantee a near perfect images but then keeps the signal totally digital until the last step of the process where image is displayed on a plasma screen, HDTV, or projector (unlike digital cable that is analog and loses quality at the two steps after your cable box). HDTV streams can also have Dolby Digital 5.1 audio tracks. When you record HDTV you are guaranteed the quality that it was originally recorded in at the stations studio and anytime you watch it after. HDTV includes 17 formats from as low as 480i stereo (Quality of a badly made DVD and 1GB/hr) to 1080i (19.4Mb/s or about 9GB/hr). 720p (Which is 1280 by 720 lines at 60fps) is best for fast action while 1080i (1920x1080 at 60 half(1920x540 lines) frames per second) is better for slow moving video.
HDTV tuners are actually very simple thing when compared to NTSC tuners. HDTV uses ATSC signals for broadcast, which is basically 1's and 0's. The 1's and 0's are MPEG-TS, very similar to what is on a DVD. Saving a HDTV or ATSC stream only requires writing it directly to disk. There is no encoding needing to be done on a computer because it was done at the TV station already. Playback though does require decoding and takes a lot of processing power to do. An ATSC signal is basically a 45Mb/s data stream. Each station can fill that stream with whatever they want. This can be as simple as one 480i stream of the TV show Nova or as much as one 1080i stream of the movie Shrek, one 720p stream of Monday Night Football, 2 480p streams of kids shows and 5 radio station audio feeds with pictures (All these on a single channel). ATSC uses what is called subchannels. A subchannel can include program guide information, multiple audio tracks per video stream, various video resolutions, pictures, audio, or closed captioning information. It's up to each station to decide what they want to fill their 45Mb/s stream with.
I should note I do have a connection to the company who produced this card. The owner of pcHDTV visited our local Linux Users Group a year before the release of the card and demo'd the first development card working under Xine and using a coke can for an antenna -- Quite funny. Two months before the release I sent him an e-mail asking about the status. I was offered one of 5 development cards in return for writing support in MythTV for it. A few weeks later MythTV had support for the card and was released before the card even shipped. Since then the HD-3000 card has been released which is a newer tuner design than the HD-2000 and the HD-3000 also supports QAM 64 and QAM 256, and s-video and composite video in. In 2005 support was added to DVB and the Linux kernel (versions 2.6.12 and newer) for the card so the only thing needed to get the card running was copying a file downloadable from pcHDTV's site to a system directory and that's it.
The first air2pc card was released in 2005 for Windows and then later support for Linux. The newest card, the HD5000, also works well with Myth for OTA but has mixed results for QAM 256, and QAM 64 only has 10% or so success rate. The cards can sometimes be found on closeout specials for much below retail.
Support for Fusion cards has been in the works since 2002, but Fusion hasn't been kind the the Linux community. When they released the HDTV5 Gold and Lite cards, Linux hackers got these versions of the card working. The Gold version includes the latest generation of HDTV tuner chips (Same as the other brands use) But the Lite version uses an older generation (Same as the HD-2000) and the tuner quality has been reported to be very, very bad. The gold version should be the only version even considered for Windows or Linux use.
It depends on how you want to use the card. All (Minus the Fusion HDTV5 Lite) work well for OTA as long as your antenna and cable are good. The pcHDTV HD-3000 is the clear leader in QAM support and is the only card that works in some areas. For support, pcHDTV is also the only company you can get support for Linux from. In general, pcHDTV is the only company that has become a real friend for Linux HDTV. They've opened everything about the card, given hardware or money to help developers improve drivers and open source projects, given money to the EFF to help fight to keep HDTV open, and are completely dedicated to providing us with something that will work with Linux. The card is the best supported card in MythTV and HDTV development for Myth is done with this card. It's by far the most used card under Linux and although I'm biast towards it, it's hard not to be given these things. If you're looking to get a card, I'll recommend the HD-3000. :).
Visit www.antennaweb.org and you can find out what TV signals you can get, how, and what direction to point your antenna. If you're using cable then visit avsforum and check their HDTV signal forum to see what non-encrypted channels are available.
Possibly. If your HD set top box supports firewire, and is on a list of supported set top boxes you can stream the HD streams off the box and MythTV and Media Portal will save these. The best way to find out is to google for your set top box and the phrase "firewire capture".
Windows does do HDTV, but MCE is terrible so you will want to use Media Portal. It's best to visit their site to find out what works best for it because it's been changing a lot.