What is all this about different ellipsoids to represent the earth you ask? Well, the main reason (apart from refined measurements from satellite networks) is that one model may be optimized from some specific part of the earths surface (such as Europe or North America) rather than the earth as a whole.

But since we are talking about ellipsoids, and only the size and not the orientation with respect to some celestial reference frame or some such, we only need to specify two parameters, the major and minor axes of the thing. As you probably know the earth is fatter at the equator than at the poles (i.e. the equatorial radius is bigger than the polar radius). A parameter called "flattening" is given by (a-b)/a. Here are some of the numbers (in meters):

Datum	Equator, a	Pole, b	(a-b)/a
Clarke 1866	6,378,206.4	6,356,583.8	1/294.98
GRS 80	6,378,137	6,356,752.3141	1/298.257222
NAD83/WGS84	6,378,137	6,356,752.3142	1/298.2572236

Both the Clarke 1866 and GRS 80 are tailored to North America; NAD83 is global.