What Can Be Photographed with Just a Point-and-Shoot Camera
Note: When I say "point-and-shoot," I assume that you know how to use your camera and can use manual settings for exposure and aperture. I also define "point-and-shoot" as NOT an SLR.
Required Equipment: Camera with manual focus, exposure, and aperture.
What Can be Photographed: The Moon
Major Limiting Factor: Camera steadiness.
The short answer is "not much." Mainly, you can photograph the moon. This is because the moon is, well, pretty bright. However, unless the moon is between first and third quarter (half-full or more), you still may need a tripod (next section) in order to properly expose the moon unless you have a fast lens (which is rare with the point-and-shoot cameras).
As far as exposures are concerned, many people say to stop down the aperture to f/8 or even f/16. I, however, completely disagree with this unless it's a focusing issue or flaring issue at low apertures. My philosophy is that you don't have a tripod and so you're trying to take as fast a picture as possible to minimize hand movement. Thus, my advice is to use the largest aperture (smallest f-number) you can, and then experiment with exposure length. And use the largest optical zoom you can (unless things have changed, digital zoom is pretty bad and you're just as well off using image editing software to blow the image up).
The Use of Manual Settings
Manual settings are a MUST for astrophotography. Your camera's auto-aperture and shutter speeds are calculated by looking at the average brightness of the entire field of view, not just the small object you're trying to photograph. Consequently, if you set a manual aperture and allow the camera to choose a shutter speed, it's going to choose a very slow shutter to try to expose the black sky "properly" and completely saturate the moon. The same is true with a short shutter speed. So you really need to select both yourself and use that scary "M" (for manual) mode setting on your camera.
Adding a Tripod
Required Equipment: Camera with manual focus, exposure, and aperture. Tripod.
What Can be Photographed: The Moon. Artificial satellites (trails). Planets as points. Pleiades.
Major Limiting Factors: Earth's rotation and limited exposure time.
A tripod opens up a few more options. With a tripod and a point-and-shoot camera, you should be able to take an exposure for up to 30 seconds, which allows you to do a few more things.
One of the first - and very often over-looked - is Earthshine from the Moon. The Sun shines light on Earth, but it also shines on the Moon. That makes day on Earth and day on the Moon, and where it's day on the Moon, it appears very bright (what we normally think of when we see the Moon). But there's another reflection that goes on: Light reflects off Earth, onto the Moon, and then back to Earth (so we can see it).
There are two differences with this light. The first is that it's much dimmer since Earth is far from an ideal reflector. The second is that the light from Earth reflects off the entire surface of the Moon that is facing Earth at that time, as opposed to just the sun-lit day side or unlit night side. This means that when the Moon is in a crescent phase, you can still see the entire disk of the moon due to the Eartshine off of it. The newer the Moon's phase (the thinner the crescent), the brighter the Eartshine will appear because there's less of the sun-lit portion showing.
And you can photograph this. It requires an exposure of a few seconds, and the Sun-lit part of the moon will be completely saturated. But, you can see it, as in the example below, which was an 8-second exposure, tripod-mounted shot at 18 mm (28.8 mm equivalent) at I think f/3.5.
Other observations that open up are satellite passes (they appear as bright streaks in a time-lapse image) soon after dusk, the Pleiades, and planets (if you don't have a long enough lens, then you can at least record any interesting conjunctions, such as Jupiter and Venus being close together in the sky, for example).
Having a Bulb Setting
Required Equipment: Camera with manual focus, exposure, and aperture; ability to set exposure length to "bulb" which means you can expose as long as you want. Tripod.
What Can be Photographed: The Moon. Artificial satellites (trails). Planets as points. Pleiades. Star Trails.
Major Limiting Factor: Earth's rotation.
Star Trails - Length
The Bulb setting allows you to take an indefinitely-long exposure (as long as your camera battery holds out and as long as the noise is low enough). The main benefit to a bulb setting at this stage is that you can take pictures of star trails. Star trails are just that -- trails of stars that are made as Earth rotates. Here is how you can figure out how long the star trails will be for a given exposure length:
So take your exposure length in seconds and divide by 240. The result will be how many degrees long your star trails will be. For example, take an exposure for 15 minutes (900 seconds) and the trails will be 3.75° long. It's that simple.
Figuring out how many pixels long is slightly more complicated. The main part is that you have to know what your field of view is. This can be calculated by first putting your calculator in RADIAN mode (not degree mode), and then:
On an APS-C sensor, the width of the detector is 22.2 mm. A full-frame is generally 35.8 mm or 36 mm, though you don't have to be too exact here. The result of this calculation will be how many degrees wide your field of view is. For example, on my Digital Rebel, a 35 mm lens (detector width is 22.2 mm) has a field of view width of 35.6°.
Next, we figure out the plate scale. This means taking the field of view and dividing by the number of pixels. My Digital Rebel has 3456 pixels across. So the plate scale is 35.6°/3456 = 0.01030 degrees per pixel.
Putting this together with the original calculation (a 15-minute exposure) that resulted in a trail 3.75° long, we can figure out how many pixels that will cover (approximately, since we're actually dealing with a curve instead of a straight line). To do this, divide the 3.75° by 0.01030 degrees per pixel. The result is that your star trail will be about 364 pixels long.
Alternatively, you can use this method in reverse to figure out how long of an exposure you can take before you start to see trails. But I will leave that as "an exercise for the reader," as the saying goes.
Star Trails - Effect of Aperture
As far as aperture is concerned, this is another case where I've often heard people say to use f/8 or so. Unlike the Moon where I whole-heartedly disagree with this advice, your aperture will give you different effects with star trails. One thing to keep in mind is that the sky glows: If you just set your camera facing up and leave it exposing overnight, it will probably be a fully saturated image when you check it in the morning. In fact, it will probably be saturated within an hour or so. As such, you need a very dark location if you're doing star trails for more than hour-long exposures.
One thing that helps this is the aperture: Stopping down (increasing the f-number) the aperture will decrease the amount of light that exits through the back of the lens and so it will decrease the glow recorded from the sky. However, it will also reduce the number of stars that are recorded because the camera detector will not have a large enough dynamic range to record them (they will now appear fainter because there's less light being recorded). So aperture is a compromise with star trails. You can either stop it down and be able to take a longer single exposure with less stars, or you can leave it open and get a whole lot of 'em, but it will need to be a shorter exposure or you'll saturate the detector.
Star Trails - Aesthetics
The trick is how you want them to look, which is done by your field of view and where you center your field of view. In the Northern Hemisphere, stars appear to orbit around the North Celestial Pole (NCP), which is marked by the star Polaris (the end of the Little Dipper). It is not a very bright star, but it is visible even with moderate light pollution. In reality, Polaris is about 3/4° away from the NCP and so it will show a trail, albeit very small. In the Southern Hemisphere, there is no bright star to mark the South Celestial Pole (SCP), though the constellation of The Southern Cross spans across it.
If you aim your camera at the pole (you only have one choice unless you're on the equator, in which case the poles are on the horizons), then the star trails will be arcs with a center at the pole. If you aim your camera away from the pole, then you will get arcs with no on-camera center. They can both produce interesting effects as shown in the next two example images:
Another issue to keep in mind with star trails is that foreground objects, such as the mountains in the top one and the trees in the bottom one can serve to make the trails more interesting.