I will start off with star trails because its the type of photo most people will already have the equipment needed to have a go at - all you need is a camera with bulb setting, a remote (or a peice of sticky tape and a coffee bean or pebble, anything that can push and hold down the shutter button), a wide angle camera lens, and a tripod or a place where you can sit the camera without it moving (fence post, pile of stones etc.... ).
The stars rotate 360 degress in 23 hours 56 minutes and just over 4 seconds or approximately 15 degrees per hour (Clockwise in the Southern Hemisphere) so when you see a star trails photo you can judge how long the shutter was open for (by looking at the angle one star has rotated around the pole) - the minimum exposure time I recommend is 30 mins (only 7.5 dregees) since there are so many stars in the night sky they will start to appear to link together to give you the effect of a circular motion but longer will make this stand out more, if you are using a digital camera the main limiting factor will be that your batteries will go flat, usually within a few hours (unless you have an external power supply) A basic exposure is ISO 200 f/4 for 30mins (so almost any lens will do this, wide angle is recommended) as long as you are in a dark sky area (with no light pollution around, and no moon) you can expose for as long as you like, the stars do not stay in the same place long enough for you to burn out the photo, but once again longer is better. There are two main ways to take a star trail photo, one is to do it with one long exposure which is great if your camera dosen't have to much camera noise, or two is to take shorter exposures (of the same lenght) and layer them together with photoshop (make sure the gap between each exposure is as short as possible - otherwise you will have missing gaps of light in your trail) With my Canon cameras I am undecided which way works best.
When using a normal camera tripod there are 3 main factors involved in how long star trails will be:
Time - the longer you expose for the longer the star trails would be.
How much you zoom in, or your field of view - the more you zoom in the narrower the field of view and hence the faster the star will pass over that field of view, giving you the appearance oflonger trails
What part of the night sky you are pointing the camera - if you are pointing towards either of the poles you can expose for longer before you notice the stars are moving, but as you get further away from them (max 90 degrees) the faster the stars will appear to move or the longer the trails will be in your photo.
There are two main effects that you can capture with star trails, one is to have the stars rotating around the poles (looking South or North) and the other is to have a scooping effect where the stars on the equator in the night sky are going straight across your frame and then the stars on either side arcing in different directions (looking West or East)
Rotation around the South Celestial Pole (SCP)
In this image we can see how much the stars have moved during the time of exposure. The Earth's rotation axis is located where the stars appear to be moving least and sadly for the Southern Hemisphere there is no bright star to mark this point. In this case the exposure is 30 minutes, ISO 320, f/2.8 taken with a Canon 1DS mkII and a 16 - 35mm lens (16mm).
The background glow is still quite bright so an aperture of f/4 - 5.6 should remove a lot of this, some of the lines that the star make have missing segments this is where a cloud has passed over and blocks the light during that part of the exposure
Star Trail photo taken from the summit of Mt John looking South, with Lake Tekapo Village on the Left and the Canterbury University 1 meter telescope dome on the right, click on the photo to download full sized (high resolution) image
16mm - f/5.6 ISO 200, 2 hour exposure, taken from a Canon 5d mkII and 16 - 35mm f/2.8 lens
Looking East from Mt John over Lake Tekapo, in this star trail photo we can see the arcing motion of the stars, the star on the left are in the northern hemsphere and the star on the right are southern hemspere. Where the stars are moving in a perfectly straight line this marks directly East and if you were to measure the angle from a level horizon this is your latitude. Photo taken with a Canon 5d mkII and a 16 - 35mm f/2.8 lens, exposure time 2 hours, ISO 200 f/6.3 - Click on photo to download full sized version
Longer startrails can be made by taking a long single exposure or by combining many shorter exposures and putting them together with software like www.startrails.de or www.starstax.net both these programmes are freeware, what is kind of interesting is that you can make growing startrail animations, heres an example using the StarStax software
Growing startrail animation made by adding images of a timelapse animation progressively together.