There are two types of low light photography essentially: snapshot style, and slow shutter style.
Almost any camera can take very nice slow shutter style night shots, but it requires learning some techniques like setting low ISO levels, using timers or remotes to trigger shutters, and using tripods or level surfaces to snap the shot.
For the other type, which sounds like the type you are looking for (snapshots - or handheld shots of indoor or poor lighting situations). One method is to use the flash - you will be limited to the flash's effective range (which on compact cameras can be just a few feet sometimes) and get well-illuminated subjects with dark backgrounds. Or you can use high ISO levels, and a wide-open aperture that has a good low light sensitivity instead of a flash. However, most compact cameras will struggle with this - high ISO values are usually very grainy and noisy on compact cameras and you lose alot of detail like hair and textures, and lenses are usually not capable of very wide aperture values required to absorb as much ambient light as possible. A few - like Fuji's F20/30/40/50 were better than their competitors in this area.
Otherwise, you'd have to look at getting a DSLR camera. These have significantly larger sensors than compact cameras, therefore allowing much higher ISO sensitivities allowing them to shoot snap-shot style in low light and still not have too much grain and noise in the shot. But often the 'kit' lenses that come packaged with the entry-level DSLRs are not suitable for this type of shooting, so alot of folks who are photography amateurs and buy themselves a DSLR because of all the marketplace hype promoting them end up taking worse photos than they did with their compacts because they don't properly understand how to set and adjust the camera for best performance, haven't learned the different style of shooting with an optical viewfinder and a much shallower depth of field on focus, and are often only using the one cheap kit lens that came with their camera.
That said, even with a cheap kit lens, a DSLR can usually outperform a compact camera in low light situations, but not by as much as you'd think - the great gain in high ISO sensitivity being offset by the several-stop loss in aperture size limiting the amount of light the lens can pull in. Ideally, for low light shooting hand-held style, you'd want a DSLR with a lens that has an aperture value of F2.0 or better (smaller F number means WIDER aperture and more light getting to the sensor, while smaller F number means smaller aperture and less light getting to the sensor.
Just remember that with the DSLR, you are committing to a larger, heavier camera designed to take various lenses as needed for each function, have lots of user-controllability, and usually at least a small knowledge of photography basics since most of the time you don't get to see how all the settings come out until AFTER you press the shutter (compact cameras are 'live view' style, meaning you are seeing on screen exactly what you are going to capture to the sensor...with DSLRs you are looking right through the lens glass, so exposure, white balance, contrast, sharpness, color, etc won't be known until you review your shots.
Hope that didn't confuse the wits out of you. In the compact arena, you don't really have any current choices that are going to be great in the situation you are asking about. With a DSLR, you should have no problem, but are buying a fairly substantial camera and to get the most out of it will probably need to brush up your photography knowledge and consider some additional lenses.