An equatorial mount, aligned to the earth's
rotational axis, is essential for anything but very superficial
astronomical observations.
It is used to mount a telescope, or other optical instrument, and differs from the altazimuth mount in that the telescope only needs to pivot around a single axis in order to compensate for the rotation of the earth. With an altazimuth mount, the observer needs to alter two axes simultaneously, and at varying speeds, to keep an object in the centre of the field of view.
The altazimuth mount is a sophisticated name for the basic twist and tilt mount fixed to the top of many cheap tripods. It allows the telescope to twist about a vertical (or near-vertical) axis, and to tilt about a (roughly) horizontal axis. For telescopes sited anywhere but the equator, this is all but useless for serious observing, and totally useless for astrophotography.
When observing a star, planet or other extraterrestrial object, the motion of the earth relative to the fixed stars causes the observed star to drift out of the field of view in a few minutes (or seconds, if observing at a high magnification). In order to compensate, the observer has to track the object by moving the telescope slowly, so that it points directly at the star, while the earth moves underneath the telescope.
The equatorial mount is designed to overcome at least half of this problem. The idea is that when the equatorial mount is properly aligned with the earth's rotational axis, the observer needs only to move the telescope about a single axis, at a constant speed, in order to compensate for the rotation of the earth.
Additionally, if a small motor (electric or clockwork) with the correct gearing is fitted on the relevant axis, the telescope can predictably track any celestial object, allowing long-exposure photographs to be taken with no risk of blurring.
There are two broad types of equatorial mount: the German EM (GEM) and the Fork EM (FEM). Both operate on the same principle, but are different in the detailed design.
The German style is heftier and a bit more complicated to build. It is usually used for smaller refracting telescopes, and amateur use, while the fork type tends to be used for larger scopes, and reflectors-especially large professional astronomical instruments.
The equatorial mount has two axes, called the Right Ascension (RA) and declination (Dec). Once the mount is properly aligned, a star can be observed by adjusting only the RA setting, leaving the Dec. setting unchanged.
To align the mount, the axis of RA pivot needs to be exactly parallel with the earth's rotational axis, (setting the dec to 90° and pointing towards the Pole Star, which has a declination of roughly 89 deg 15 mins is good enough for most observations)There is an excellent description of how to align an equatorial mount here:
http://www.vgernet.net/chm027/align.html