Actually, it's not nearly as complicated as it might seem. The key to this lies mostly in the video card included in the computer in question. Most of the better video cards (the ones that aren't bottom of the line, bottom of the barrel) will have at least two, possibly three options for output. Many video cards offer a standard 15 pin VGA connection, an HD DVI (Digital Video Interface) port and possibly an S-VIDEO output port. Some of the higher end cards have eliminated the old standby VGA port and include multiple DVI ports.
The trick to connecting a TV to your video card is to know what both devices are capable of. Most TVs - especially the modern HD sets - have multiple ports - some have VGA, S-Video, Composite, DVI, HDMI, Cable coax, etc... Many older sets have fewer ports - maybe only a coax port for cable, a composite RCA port set for a DVD player and maybe an S-Video port for whatever. You'll need to figure out exactly what you're TV's got and has available. From there, you can work backwards to figure out what kind of video card your PC is going to need.
Setting it all up...
You can actually buy an off the shelf Media Center PC that comes with Vista Home Premium or Ultimate - you'll need one of these two versions if you're buying a new computer as they are the only ones that come with the modern version of Windows Media Center for Vista. Most of these computers also come with a remote control - so you can access the box from the comfort of your couch.
If you've already got a computer collecting dust, you could also buy a few upgrade components (like a video card, a remote control interface, etc...) as needed to make your own. You can simply use PowerDVD, WinDVD, Windows Media Player or other programs that can access files on your hard drive OR, if you really want to get deep into it, you can get a tuner card such as a Hauppauge WinTV card and a copy of BeyondTV (or something similar) and turn the media center computer into a DVR as well. Hauppauge offers both Standard Def and High Def models with single or dual tuners.
And, of course, you'll need at least one, if not two LARGE hard drives - preferably 1 Terabyte or larger (each). That way, you're not likely to run out of room unless you've got a HUGE collection. Remember, standard DVDs in their native format take up anywhere from 4 GB to 8 GB in size. If you want to keep ALL of the stuff on the DVD - i.e. the menus and bonus features, etc... you'll probably want to keep the DVD in it's native format instead of reducing it to some other format. Bear in mind that converting to other formats may reduce the quality of the video and audio noticably.
So... In short, it IS possible, and it IS easy enough to do with the right components and software.
Now comes the sticky wicket... You mentioned you would like to put all your media on your hard drive and make the DVD cases vanish from your shelves...
While you CAN, IN THEORY, do all of this, it isn't legal if you purchase (or rent) a copy of a movie, rip the contents of the DVD onto your hard drive and then dispose of/return it. The MPAA frowns heavily on the practice and should they come across your monster system, fully loaded with movies, CDs and the like, they would quite literally soil their shorts - that is, before they sent in their crack legal team to sue you into oblivion.
Note: SOME people claim that as part of "fair use", you have a right to back up a copy of your media, in case some disaster should happen... Of course, you still need a copy of the original media. Obviously, the MPAA and the RIAA would tend to vehemently disagree with this point of view. You would likely still get sued, but, you'd stand a better chance of having the case thrown out if you have the original media in it's original packaging. So much for clearing the shelves...
On a more positive note... I've found that BeyondTV and a tuner card makes for a pretty darn good DVR.. Better, in many ways, than TIVO or some of the other DVRs on the market. BeyondTV (www.snapstream.com) offers up a free programming guide in addition to the software you get to record your shows and skip the gawd awful commercials. And the best part of it all - it does NOT automatically delete content, unless you're running out of room and you've flagged a show as disposable in a low free space situation. You can record an entire season of a given show and have your own commercial free full season marathon at your leisure with the ability to pause, rewind and stop should you get an overdose of that show.
There is one important consideration as far as tuner cards are concerned. Most of your older stock tuners will connect to standard cable and will function without a cable box. This will generally give you channels 2 - 125 (or channel 99 on many cable systems). To get higher channels - especially HD channels, you'll need a tuner card that can handle the job AND you will probably require a "CableCard" from your provider. You'll want to check out the availability with your cable provider before you invest in the tuner card. Without a CableCard, you will require an actual cable box and be limited to only recording ONE program on ONE channel at a given time.
The bottom line: Yes, a PC can make a very potent component for your entertainment center. But keep in mind, there are still some legal issues that haven't been 100% resolved as yet.
As long as your media center box is NOT connected to the Internet and you're NOT sharing any media, and you're not bragging about the awesome new system you've put together in public, you should be fine.
Your mileage will likely vary...