Star trail photos use a very slow shutter speed to blur the movement of the stars across the night sky; this is the same technique as waterfall photography. Both genres call for a tripod and some patience, but the long exposures and lighting conditions bring up extra difficulties. While a proper exposure is necessary just like any other type of photography, the mechanics of how one is obtained are a bit different in this case.
How long the exposure needs to be depends on the lens you're using. A wide angle lens shrinks things in the frame, including the stars and their movement. A longer lens magnifies the trail.
In general, 600 / FL gives the number of seconds before the stars will begin to streak; also known as "the rule of 600." With a 50 mm lens, a 12 second exposure will give pinpoint stars, while longer exposures lead to more and more noticable trails. It takes much longer to get the type of long streaks most people are used to.
The image at right ( Zion ) was an eight minute exposure at 26 mm. Below ( Pike's Place ) is about a fifteen minute exposure at 19 mm. Ultimately the length of the exposure has the most bearing on the length of the trails in the photo. Plan on a five to fifteen minute exposure.
The aperture you choose depends on the focal length, as well as all the usual factors. Because the stars are moving, and more quickly the longer the lens, your shutter speed doesn't have much affect on how bright the stars come out. This is controlled with the aperture.
Without much light pollution in the sky, a wide angle lens ( 35 mm and down ) will record clear trails at f/8. In Seattle I try to shoot at f/5.6 for most star trail photography, to make sure dim stars are picked up.
Most digital cameras don't tolerate long exposures like film.
Ugly specs ( dark current noise and hot pixels ) show up across the image, in some cases it will be accompanied by a colorful "cloud" where part of the camera heated and leaked current. My D60 would show a magenta stripe down the right side of an image after about ten minutes of exposure, that would get stronger as time went by. In fact many digitals are limited to shooting in about the 30 second range.
This can be overcome with software and shooting technique.