A long exposure, and a gorillapod to keep the camera completely stable.

Well, first of all, it's difficult to shoot a moving object at night - the camera (this is especially true for a point and shoot) takes longer to respond because it needs more time to get the focus (autofocus is pretty much helpless in the dark) and exposure right, and the exposure time is longer as well, so by the time the picture is finally "shot", the object is long gone, as you can see from the picture.

Secondly, you need to ask yourself what it is that you want to shoot. In addition to the Moon and the taxi, you need to pay attention to other things inside the shot as well: right now, there's a lot of stuff that's pure noise, like the traffic lights. It's not obvious at all that you wanted to take a picture of the Moon, as it's in the background, blocked by all those other things.

The first thing I can see is that the aperture was f/3.5 and I'm guessing was focused on the cab (which is invisible other than its bright lights because of the long exposure). To get sharper focus in this situation using the tools you have, you might want to try setting the camera into landscape mode. This basically instructs the camera to use a smaller aperture (higher #: f/8, f/11) and favor a prolonged exposure over opening the aperture. With that more closed aperture, the depth of your focus increases.

Do you understand how the balance between aperture and shutter speed works? Since I'm bored, I'll just type it all out in case you don't. Aperture = the size of the hole letting light in: shutter speed = the length of time the hole is open. So the smaller the hole, the longer the time needs to be to let the same amount of light in. (It can help to think of light like water. And also, just remember that you're dealing with fractions: so bigger numbers are actually smaller.) In your camera's settings, portrait mode favors a larger hole. If you stick with the water analogy... a larger area pool will be shallower to hold the same amount of water. So... Shallow focus. Landscape mode favors a smaller hole that will be like a well... Deep focus!*

Were you trying to catch an image of the taxi (something other than its lights)? If so, you'd probably want to use flash: a lot of cameras will have a nighttime-portrait-with-landscape mode that uses a quick burst of flash to catch your foreground subject while using a longer shutter to capture the un-flash-able background.

I wonder if I could say that the taxi cab has a finite amount of light water that it's splashing back at you... So as you extend the shutter length you're spreading the light water out, and getting less and less of it at each point in time/space. Eventually you spread it out so thin that it's practically invisible. The way to overcome that is to pump in some more light water (i.e. flash!).

The lights and moon turn into big colored blotches because they have soooooo much light water coming from a concentrated space/time that they are overflowing themselves as you keep the shutter open for longer. (That's not incredibly accurate, but I think it generally works based on the end result.)

Sooo... The lights on the cab are thin, sharp lines because we are spreading that large amount of light water out over time/space. Also, the cab is in our shallow plane of focus.

Maybe this all sounds totally crazy. It's been a long week, full of sickness.

Is there something particular you were going for? Maybe there's better specific advice to be given.

*This one is really unrelated to everything else. Hmmm.

The way to force a point-and-shoot to use a shorter exposure is to set the ISO up to at least 400 and reduce the exposure compensation to -2. Both of these will reduce picture quality, but they increase your chance of the picture being visibly of something.

Remember that when the shutter speed is very slow, even your pulse produces enough movement of the camera to blur a shot. A solid surface or a tripod is completely necessary for this kind of picture, even for experts.

This is all great advice, that I will take into account the next time I try for a night photo.

Thank you all very much!