Camera Advice

I do not think that getting a DSLR is the answer to many peoples problems. For most people a good Bridge Camera is more than is required & a decent compact will be fine.

I would be like a Motor Forum advising everyone to buy Sports Cars.

The problem here is that on 'Night Mode' the program will probably be choosing longer shutter speeds & so there is blur. Some sort of stability will probably solve the problem. An entry level DSLR will still have the same issues. If the OP wants to progress their photography furtther then some better equipment should be considered.

The pictures are blurred by motion - even the motion when you press the trigger. Try lying the camera on something (a rail, a bench, holding it against something (wall, sign post, lamp) or to test it get some of the cheapest tripods e.g. this one.
The image quality in low light conditions for compact cameras is very similar - bad. The improvement would be going for camera with larger sensor - digital SLR as was suggested by ricky91t.

I think much (if not all) of what is being suggested above has merit even though some of it is conflicting.

OP - to get a handle on what the night problem is, try the following;

find a night / low light scene you want to take.
find a steady surface to sit your camera on.
set timer on your camera.
press the shutter release and wait for the camera to do its thing.

Does this give you a clear or blur'd image?

If clear, then you are into steadiness issues and the tripod option is a good way to steady your camera before taking a shot.

If blur'd then it may be that your camera is hopeless at focusing in low light (i had one of these which was fine during the day but night or low light was disastrous). You could try manual focus and low aperture if your camera allows for manual control but in my experience a camera that doesn't like low light is probably not going to improve much.

If its hopeless in low light then you may as well be pee'in again the wind as trying to get a nice shot in such conditions (neither a pleasant experience ). So if the latter, then i'd opt for a camera replacement (assuming its something that you'd really want to do) - you may still need a tripod as even the most minute movement (including your own breathing) can upset the image.

The one that ricky suggests looks on the face of it to be smashing value - darn fine camera for the money. This still mightn't resolve your problem but it will be far more capable - to get what you are anticipating may involve getting a better lens for the camera which is more suited to low light situations which you could spend as much as the camera to get but it would be a starting point.

Hope that helps.

You see - this is the problem with asking advice on a forum - you're likely to get as many conflicting responses as you get responses :P

Basically it depends on where you want to go and what you want to do with your camera. If you're simply wondering how to cut out blur and you'd then be very happy with your shots (and for a whole lot of people this would be what they're asking) then I'd 100% go with what an cat dubh said; you may already have a perfectly capable camera that just needs a tripod or using it with the timer to be happy with your shots. Can you borrow a tripod somewhere? If the camera is on night mode then its keeping the shutter open a lot longer to compensate for low light, and the slightest movement can ruin a shot. Try it out and see if this solves your issue.

If you'd like better shots and want something that's flexible and not too expensive, then a bridge camera is probably best. They have huge zoom ranges and you never have to worry about changing (or buying!) lenses. Some of the newer models have fantastic image quality.

If you're thinking of getting into creative, higher level photography then an entry level DSLR might suit you best. They have great control over what you capture, you generally get a better quality of image (although less money doesn't always mean less quality) and they're more or less infinitely upgradeable - you might stick with teh kit lens for a bit and then get a few better ones and then get a better body and so on - BUT.... This is by far te most expensive option. The kit lens on the above mentioned model is fine enough, but the zoom capabilities on a bridge are a LOT better. Basically, the bridge will give you a good all round camera straight away for your bucks but the DSLR will give you a better long term investment if you're serious about photography.

If you try the tripod and timer test and its still just not cutting it for you, I suggest you go into a few specialist camera shops (not the likes of Dixons et al), try some different models in your hand and get a second and third and fourth opinion on what you really need. As Cabansail said, there's no point buying a ferrari when an audi estate is what you really need*

*I know NOTHING about cars - is there an A8 estate?