The camera's brand isn't going to make much of a difference. What will make a difference is your method of taking a photo and the features of the camera.
Personally, I hate using a flash - it just throws all the colours out of whack and looks awful. Without a flash, though, your shutter will have to stay open for longer, which means that you're going to have to keep the camera still, preferably on a tripod but any stable surface will do.
Your typical point-and-shoot camera will do the thinking for you, which would be well and good if it could actually make the right decisions. You're much better off getting a camera that will let you manually adjust your settings like aperture size and shutter speed, and once you get the hang of it, you'll find that you get much better results than with a point-and-shoot.
These shots were taken at night with no flash using a Canon Powershot S1IS (which has since been superceded with the S3IS):


The S1IS isn't the most whiz-bang camera around, but if you learn about things like lighting conditions, you can get some pretty decent results. Here are some samples of different kinds of lighting that I posted in another thread recently:
Natural sunlight:

Overcast:

Light box:

Red tungsten light:

Bottom line: get a camera with manual settings, as well as a tripod.