¿Qué significan los términos geoide, elipsoide, esferoide y datum y cómo se relacionan?

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The geoid is defined as the surface of the earth's gravity field, which approximates mean sea level. It is perpendicular to the direction of the force of gravity. Since the mass of the Earth is not uniform at all points, the magnitude of gravity varies, and the shape of the geoid is irregular.

Click on the link below to access a website maintained by the National Oceanographic & Atmospheric Administration (NOAA). The website has links to images showing interpretations of the geoid under North America.

NOAA Geoid Index

To simplify the model, various spheroids or ellipsoids have been devised. These terms are used interchangeably. For the remainder of this article, the term spheroid will be used.

A spheroid is a three-dimensional shape created from a two-dimensional ellipse. The ellipse is an oval, with a major axis (the longer axis), and a minor axis (the shorter axis). If you rotate the ellipse around one of its axes, the shape of the rotated figure is a spheroid.

Note:
The semi-major axis is half the length of the major axis.  The semi-minor axis is half the length of the minor axis.

For the earth, the semi-major axis is the radius from the center of the earth to the equator, while the semi-minor axis is the radius from the center of the earth to the pole. A particular spheroid is distinguished from another by the lengths of the semi-major and semi-minor axes. For example, compare the Clarke 1866 spheroid with the GRS 1980 spheroid and the WGS 1984 spheroid, based on the measurements (in meters) below.

Clarke_1866  6378206.4  6356583.8
GRS_1980     6378137    6356752.31414
WGS_1984     6378137    6356752.31424518 

A particular spheroid can be selected for use in a specific geographic area, because that particular spheroid does an exceptionally good job of mimicking the geoid for that part of the world. For North America, the spheroid of choice is GRS 1980, on which the North American Datum 1983 (NAD83) is based.

A datum is built on top of the selected spheroid, and can incorporate local variations in elevation. With the spheroid, the rotation of the ellipse creates a totally smooth surface across the world. Since this doesn't reflect reality very well, a local datum permits local variations in elevation to be incorporated.

The underlying datum and spheroid to which coordinates for a dataset are connected can change the coordinate values. An example using coordinates within the city of Bellingham, Washington follows.

DATUM      X-Coordinate          Y-Coordinate 

NAD_1927  -122.466903686523  48.7440490722656 
NAD_1983  -122.46818353793   48.7438798543649 
WGS_1984  -122.46818353793   48.7438798534299 

The X-Coordinate is the measurement of the angle from the Prime Meridian at Greenwich, England, to the center of the earth, then west to the longitude of Bellingham, Washington. The Y-Coordinate is the measurement of the angle formed from the equator to the center of the earth, then north to the latitude of Bellingham, Washington.
If the surface of the earth, at Bellingham is bulged out, the angular measurements in decimal degrees from Greenwich and the equator will become slightly larger. If the surface at Bellingham is lowered, the angles will become slightly smaller. This is how the coordinates change based on the datum.