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Powershot S1IS (which has since been superceded with the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.