NAD83, the North American Datum 1983, is based on the spheroid GRS80, while NAD27, the North American Datum 1927, is based on the spheroid Clarke1866.
The spheroids are different representations of the shape of the earth; thus, they have different radii, diameters, and circumferences build into them. The mathematics that express the relationship between points on the earth's surface are based on the angular relationships between the points and the center of the earth. You can change the distance from the points to the center of the earth, as well as the angular relationship, to produce different coordinates for the same points.
For a more thorough discussion of this topic, see "spheroids, vs. spheres" in ArcDoc.