Coordinate systems provide a basis for identifying locations on the earth's surface. Historically, many coordinate systems, have been devised, with the oldest ones assuming a flat earth, then later on a perfect sphere. Today's geographic coordinate systems are now based on a cartographic spheroid (an ellipse rotated about its axis) to more closely represent the earth's true shape and surface. Additionally, map projection coordinate systems have also been developed to determine how the curved surface of the earth is depicted on a flat map.
A modern geographic coordinate system, uses latitude and longitude to describe a location. The coordinate system is based on a origin, the intersection of a chosen prime meridian and equator. The coordinate system can pinpoint exact locations, but are not necessarily idea for computing units of measure, since the distance represented by a degree of longitude changes depending on latitude - meridians are closer together far from the equator, converging to a point at the poles. Additionally, the earth is not a perfect spheroid. The South Pole is actually closer to the equator than the North Pole, and elevations on the surface are varied. Because of these conditions, the center of a coordinate system based on a spheroid is slightly adjusted so that part of the spheroid lines up remarkably well with a portion of the earth's surface. This adjustment is different, depending on what region of the surface is being mapped, so many different datums are used throughout the world.
A second cartographic problem is that flat maps are used to portray an area that is a curved surface in reality. Because of this, a map projection has to be defined when drawing a map. Some common projections can be though of as wrapping a cylinder of paper around the earth so that it is tangent at the equator, or perhaps attaching a plane sheet of paper to the earth at one single point. Numerous map projections have been created, but all map projections will include some distortion. However, different projections allocate the distortion to certain parts of the earth while minimizing it in another, so choosing an appropriate projection for a particular map is important.
In GIS, specifying or defining the coordinate system and related information for a set of data is important in producing valid analysis results, because inaccurate locations and measurements can diminish the accuracy of calculated values. GIS software usually includes extensive support for a wide variety of standard coordinate systems, and contains tools for conversions and reprojections between systems. See the Specifying a Coordinate System or Reprojecting tutorials for more information.
More coordinate system information is available through the built-in ArcGIS documentation under the heading "Map Projections and Coordinate Systems", as well as the following articles: