You need two things.
A digital camera with manual controls and a tripod.
Low light photography will almost always require that you use slow shutter speeds. You should not hand hold a camera when shooting at shutter speeds slower than 1/60th of a second.
Manual controls such as "Shutter priority" and "aperture priority" give you better control of exposure settings.
For under $300 I would start by looking at the Canon A610.
It seems like it's about time to update my Canon PowerShot A20 that I bought almost 3 years ago.
Well, now that I've had plenty of experience with it, I realize what I want in my new camera is the ability to take great night pictures. The ones taking with my A20 weren't bad, but very hard to control it from getting blurry, which is why I would like one with more manual options like aperture and shutter speed settings.
Does anybody have any favorites or suggestions for this category? I want to take the perfect night shots. Price range is about $300 or higher - I'll look into all the models mentioned anyways as I like to do that kind of research before buying anything.
Other considerations include (good battery life - and if possible, cheap battery parts).