Most people here already know about the X68000, but for those that don't, it's a 16 bit Japanese computer with a ton of perfect ports of late 80's and early 90's arcade games. Its graphical capabilities are superior to the Genesis or SNES, and even the Amiga. It also has superior versions of some console games like Galaga '88, Thunderforce II, Granada, Phalanx, and Sol Feace. The Xenocide files has an incomplete list of X68000 shmups that'll give you an idea of the impressive library this system has. Lots and lots of shmups, especially from Konami. The original X68000 was sold with Gradius included, actually.
The Hardware:
I bought the first model X68000 (because it was cheaper), which is grey rather than black like the later models. Everybody's seen pictures of the thing, it basically looks like a miniature Cray. Dual 5 inch floppy drives, RGB output, two MSX compatible joystick ports. There's quite a few other ports on the thing, but none are of any interest if you're just using the X68000 to play video games. There is a SASI port on the back you can hook a hard disk up to, which might be useful since some games span 5 or 6 floppies.
The X68000 apparently uses two video resolutions, 15khz (Low res) and 24khz (high res). Many of us have 15khz monitors for use with arcade boards or video game consoles, but a monitor that can do 24khz is a bit more unusual. Instead of buying an X68000 monitor, which would have cost a small fortune to ship from Japan, I used a NEC Multisync II monitor I already had sitting around for my Amiga and FM Towns. It handles both frequencies just fine, but you need an adapter since they use different connectors. Most Japanese computers use the same pinout for their RGB video connector, so I could use the same video cable I have for my FM Towns.
Most of the games I played run in high resolution mode. The video is nice and sharp, like you're playing it on a VGA monitor. The only problem is that the picture doesn't span the entire screen horizontally. It doesn't matter for vertical games, but for horizontal ones it means you're only using about 90% of the screen to play on. I don't know if it's like that on all monitors or just the Multisync II. Most games have an option to run in 15khz mode, which gives you that added bit of authenticity since most arcade games use 15khz monitors. Plus the video takes up the whole screen.
The X68000 seems quite capable of perfectly emulating sound effects and music of the original arcade games, too. There's nothing else noteworthy about it, except that it can sound like your average early 90's arcade game. It's got built in speakers, but you can also connect external ones via the RGB connector. There's a volume dial on the front, and also a standard headphone jack.
Like I mentioned before, the X68000 uses 5.25 inch floppy disks. Most shmups seem limited to one or two disks, but RPGs and the like often span 6 or 7 disks. Since it uses disks rather than cartridges, there are load times when you want to start a game up. For Detana Twinbee, which is two disks, it takes about 45 seconds for the game to start up, which isn't bad. No disk swapping either, since the system has two drives built in. Most games have load times in between levels, too. But of the games I played, it's only a couple seconds. The load times are just minor nuisances, anybody who's used to Playstation or Saturn load times probably won't mind it a bit. But if you just did something stupid in a game and want to start over, it can get a bit annoying to sit through the load time again.
As for joysticks, any old MSX compatible stick will do. There are two ports, so you can play two player games. The first port is on the front for easy access, and the second port is on the back for some reason. The only downside is that MSX controllers only have two buttons. There's no start or select button to pause the game with, either! Detana Twinbee lets you pause by hitting the escape key on the keyboard, I don't know if that holds true for all X68000 games or not.
So basically you have a computer, released in 1987, that blows away everything else that was available at the time. It's not as powerful as a Saturn or Playstation, but it was released a good 7 years earlier. A lot of people say that if Sharp had released the X68000 in the US, it would have dominated the Atari ST and Amiga. What they don't realise is that all this computing power came at a high price. The X68000 cost around $4,000 when it came out! I believe an Amiga 500 from that era sold for around $600. The X68000 had many more features than an Amiga 500, though, so maybe they could have cut out a lot of unnecessary features for a gaming system (like the hard drive and video input control stuff). If they could have brought the system in around a thousand dollars, maybe it could have competed.
Software:
(note: I've only played a handful of games, so these are just preliminary observations)
What can I say? They're arcade perfect ports. If you've played the arcade version, you've played the X68000. Vertically oriented shmups don't have a TATE mode (at least not the ones I've played), but only a little from the top and bottom of the screen is chopped off. Some of the games are so accurate that you have to hit the F1 key on the keyboard to simulate inserting a coin! The downside of this means that there's no extras. The ones I've played don't have different difficulty modes and Gradius doesn't even give you continues.
From what I've seen, X68000 games are very nicely packaged. Detana Twinbee comes in a nice plastic box. Inside were disks (which have nice colorful labels), a nicely illustrated manual, a small catalog of Konami soundtrack CDs, AND a booklet of Twinbee themed postcards! It only cost me 25 dollars, too (not including shipping&handling).
It's hard to believe that some lucky kids in Japan were playing these perfect ports back when all we had was the NES and Turbografx-16. It's probably the most powerful 16 bit system out the for a shmupper next to the Neo Geo. But the question is, how does it hold up today? In this day and age of MAME, are perfect arcade ports of late 80's/early 90's games really that important? Plus many of the more popular arcade ports are available on the Saturn or Playstation. You can get Gradius 1 and 2, Salamander, Detana Twinbee, Imagefight, Parodius, etc on the PSX/Saturn. However, some of the ports of older arcade games are the best versions out there. I don't know of any other arcade perfect port of the original Twinbee, for instance.
It just doesn't seem like there are many outstanding exclusives for the thing. There's no equivalent of Compile on the MSX, churning out classic games. Nor do the Konami games seem to contain the extra levels and whatnot the MSX versions do. There are quite a few exclusives for the system that nobody's ever heard of, and while the few I've tried have been good, none are good enough that every shmupper ought to play them. Gradius fans might want an X68000 just for Nemesis 90, the port of Gradius 2 on the MSX, though. I don't think the X68000 falls in the category of essential shmupping machines like the Saturn, Turbografx-16, or Playstation 1/2.
Overall, it's a damn fine system. It's a fantastic shmupping machine. If you buy one, you'll get more than your money's worth out of it. It's a computer with a huge library of arcade perfect ports, and the games themselves are packaged better than most console games are. It's almost worth buying them just for the extras. But do you need an X68000? No. If you have an import-capable Playstation or Saturn, you can already play arcade perfect ports of most of these games. Hell, if you have a PC running MAME you can already play arcade perfect ports of most of these games. With both the X68000 and MAME, you're sitting in front of a computer monitor rather than a TV (although the X68000 can do the native resolutions whereas the PC can't without special hardware). If you want a Japanese computer, I'd probably recommend getting a MSX 2 first. There's more exclusives and special versions for that system worth hunting it down for, and it's cheaper. But if you've got a wad of cash burning a hole in your pocket, the X68000 will give you a lot of bang for your buck, plus infinite bragging rights

Summary:
+ Arcade perfect ports of tons of shmups, especially from Konami
- Most of the arcade perfect ports are also available for the Saturn, Playstation or Playstation 2, or at least MAME
+ Superior versions of some Turbografx 16, Genesis and SNES games.
- Very few good games developed exclusively for the X68000
+ High resolution RGB video
- Can't play on TV, no TATE mode
+ Most shmups are cheap and common
- The X68000 itself is quite expensive
+ Games come in nice packages with lots of extras
- Load times
+ Lots of cheap, easy to find MSX controllers available for it
- Have to use keyboard to pause
+ Owning one makes you uber-elite
- It's so obscure that you can't brag about owning one, because nobody's ever heard of it!
Notes to potential buyers:
* The thing is big and heavy, so it'll cost you at least 100 bucks to ship one to the US. And that's without the monitor.
* You'll probably want to make sure you get a keyboard and mouse with it. You need a mouse to start some games (like Gradius), and a keyboard to pause and start some games like Parodius. Most games seem to work fine with just a joystick, but better be on the safe side.
* Some games require 2 megabytes of RAM. Some X68000s, specifically the early models like the one I got, only came with 1 megabyte of RAM. I have no idea if the previous owner upgraded to two megs, or how to tell how much RAM I have, or how to install more RAM, or where to buy more RAM.
* Older X68000s use a SASI interface for hard drives, while newer ones use SCSI. You'll have a much easier time finding a SCSI disk than a SASI one.
X68000 vs FM Towns:
Which one of these two Japanese computers are better from a shmuppers point of view? The FM Towns is the more powerful of the two, no question. This is no suprise since it's a newer machine. But while the FM Towns only has a handful of shmups, the X68000 has slews of them. On top of that, the FM Towns shmups I've played haven't been that playable. You can read about it in my FM Towns thread, but there are screen size issues that make Image Fight much less playable, and Rayxanber just plain sucks. And to make things worse, most FM Towns shmups are rare and expensive, whereas X6800 shmups are cheap and pleantiful for the most part. The FM Towns might be cheaper and more powerful, but you're only getting access to a handful of games, and the only noteworthy one is Tatsujin Ou. Most FM Towns shmups are available for the X68000, too, like Image Fight and Kyukyoku Tiger. The Towns does have a pause button on the controller, though, whereas the X68000 doesn't. Don't bother with the Towns unless you're a collector (like me) or insane (also like me).