One easy way to make the camera follow the stars is to mount it piggy-back on a motorised telescope. (This is not the cheapest method - search the web for "barn door tracker" to see a more basic mechanism.) Some quite inexpensive motorised telescopes are now available and here is the author's with camera mounted on a bracket, looking parallel to the telescope:
The telescope is a Meade ETX 125mm Maksutov-Cassegrain reflector with GO-TO controller (select the coordinates of a star or other object and the instrument drives to point at it). The camera is connected by a USB cable to a lap-top PC. Canon software (free with the camera) runs on the lap-top to fully control the camera and capture images directly to the PC at regular intervals.
It must be admitted that the drives of lower-cost telescopes like this are not designed to cope with the weight of a digital SLR camera - hence the home-made counter-balance weight which can be seen at the top left of the photo. Users of compact digital cameras will be better off in this respect.
In this configuration the telescope is used purely for guiding. It rotates about an axis which is parallel to the Earth's axis. The base is home-made and angled for the latitude of the author's home so that polar alignment can be achieved. This tilted-axis arrangement is called an equatorial mount. The drive goes at sidereal (ie, star) rate, which means that one complete rotation through 360° would take 4 minutes less than 24 hours. That takes into account the fact that the Earth is also moving along its orbit around the Sun.
Without the home-made base converting to an equatorial configuration, the telescope shown above has a vertical and a horizontal axis. Placed on a flat solid base, that constitutes an altazimuth mount (short for altitude-azimuth, for the two axes). The telescope's electronics can be used in either equatorial or altazimuth mode. In the equatorial case the drive is simpler because it is about a single axis at a fixed rate. In the altazimuth configuration the electronics works out a changing ratio for driving each axis so that the telescope still follows the sky. That is fine for the visual observer but not so good for photography because although the aiming direction is correct, the field of view slowly rotates.
In the equatorial configuration, provided the drive is accurate enough, a series of photographs in piggy-back mode can then simply be averaged to get a result with less noise. In the altazimuth case that will not work because of the field rotation.
A principal reason for developing GRIP was to be able to align and average images automatically even in the presence of the field rotation due to an altazimuth mount. This has been achieved - the astro-process option on the batch menu does exactly that. In fact it goes much further and enables a series of photos from a fixed camera to be aligned and averaged automatically. GRIP finds the brightest stars in each photo and recognises the shapes formed by their connecting lines, from one frame of the series to the next. It is then able to map the photos accurately on top of each other and combine them.