To take photographs of star trails, you need a manual camera with any kind of film, a tripod, and a clear moonless night, or moonless part of the sky.Camera
The camera should have a shutter that stays open without the battery, because otherwise the battery will get used up, and there'd be no way to close the shutter.Film
Color film is preferable, because you will be able to see the colors of stars that the human eye can't pick up. Print film shows the most star trails, but slide fims shows them with more contrast. The speed film you use depends on how dark you want the sky to be. If you use a fast film (like ASA 1000) you will need to take a shorter exposure, or stop down the aperture more to keep a dark sky.Lens
Different sources suggest different lenses to use, but I would suggest a telephoto lens, because it makes the star trails appear bigger, and picks up fainter stars. However, if you are trying to capture the circle of stars around polaris, a standard or wide angle lens will show more.Aperture
The aperture is how open your lens is. For exposures of an hour or less, I would suggest trying a wide open aperture. For all night exposures, or in a place with a lot of light pollution, you will want to keep the sky looking dark by stopping down. The aperture does not effect the brightness of stars, because they are points of light, and do not expose the whole film (like the sky does).Exposure
It takes about 30 seconds for a star trail to show up in a photograph. For overnight exposures, make sure the tripod is sturdy, because any movement will make a wiggle in star trails. Other things to be careful of are frost forming on your lens, car headlights and flashlights, and places in the sky where many airplanes fly by.