Currently, the mystery dimension is entitled "The Void" and is accessable while in Creative mode, by flying off the top of the world. The terrain shape is reminiscent of the Nether, or that of the Aether mod. A semi-reliable way to reach it in survival might be by using a TNT cannon, to launch yourself far enough to land in The Void. It is completely unfinished and will be completely changed, and currently features mobs and giant land formations. For now, if you're so inclined, you can explore it, but it is of note that there are currently no dungeons. The Void begins at Overworld height 260, and, if access in survival is wished, the player must make an incredibly high block tower, and then use a jumping ability to reach it. however, this will usually result in the player just falling back into the overworld, usually not over their tower, and falling to their death.
The Void is, in fact, reachable and playable in Survival. Build a column, as described above. This will be easier if you make a very large platform at the top, to land on, as it will take several leaps. Or a bed, so you can respawn near the top. Set a bucket of water in the first slot on your hotbar, and leap into the mystery dimension, looking straight up. As soon as it loads, attempt to place the water. Be careful not to scoop it back up, as that can be disappointing. If the roof above you is too high to reach, you'll have to extend the platform in some direction and try again. If there's no roof above you, build another column elsewhere, as this is not an acceptable location. Once you get the water to stick, you'll have to leap several times, letting it slowly fall. Then leap into the dimension where the water has fallen, and quickly press "Space" to swim upwards (holding space the whole time will not work- you have to hit it after the dimension loads). Rangers may have an easier time climbing up. Once there, you can make a more formal portal by dropping 5x5 column of water and building a platform in the overworld directly below it.
Areas and biomes still work in the dimension, though biomes are subject to being removed or changed.
As of 0.5.3, the Area system gets bugged when changing dimensions in SMP, so it's best to avoid the mystery dimension when on a server. If you do travel there, the bug may be fixed by relogging.
Frizziliscollaborating with SMP to entirely retexture it. Expect something very, very awesome in the near future.