You can take night photos easily and you don't even need a digital camera that allows for the exposure to be controlled manually (although it can help a lot). I will assume, however, that your camera has an auto-timer (i.e. it can take a picture of you automatically within a few seconds after you start the timer).
Here is what you need to do:
1. Grab yourself one of those small pocketsize tripods (they are a couple inches tall). These come in handy for a lot of things. You can find them at most camera stores for about $10-15.
2. Screw the mini tripod into your camera and setup your shot the way you want it. Be sure to place the tripod on a solid surface. I have used railings, trashcans, and other surfaces for this. In a pinch, some of these mini tripods come with velcro that you can use to attach it to a pole or other such surfaces.
3. TURN OFF YOUR FLASH! This is important! If the flash fires, your shot won't be what you thought and will most likely be way too dark.
4. Set your auto-timer and let go of the camera. The picture may take a few seconds to expose before it is complete (this is due to the darkness). Do not touch the camera until you know the camera has done its job of taking the picture. Most cameras beep at the start and/or end of an exposure to indicate when it's safe to move/jiggle the camera.
5. The camera should snap a picture and it should be perfect. If it's not to your liking, just mess around with your manual settings and tweak until you get the desired look. Usually increasing the exposure time will do the trick. However, most modern digital cameras will sense that they aren't receiving enough light when the exposure occurs and will expose long enough to get you the shot. Your milage may vary.
The tripod is important because it will help make sure the camera doesn't shake. With nighttime exposures, most humans aren't steady enough to hold the camera perfectly still long enough for the photo to come out without blurring. I love the mini tripod because I can stick it in my pocket or small camera bag and pull it out for unique situations (i.e. velcroing it to the railing inside Mission:Space to get a nice shot).
I have a few shots of the castle at night on our site at:
http://www.wdwpix.com/MK,MK2003C.album
I have successfully taken night shots with point-n-shoot (Canon Elph/G2/G3) as well as SLR (Canon 10/20D) cameras, so you should be able to succesfully do it with just about any modern digital camera as long as it has an auto timer or the ability to be controlled remotely via a small remote control or remote switch. For most people, the auto-timer is a no brainer choice since it's built in and easy to use.