Like other subjects, cartography is not short of jargon. Here's an attempt to summarise the main terms. As stated on the Where Am I page, the earth is not a perfect sphere but an ellipsoid, so how you define this is your reference ellipsoid. How far you are from the earth's centre will depend on where you are on the surface and the shape of the ellipsoid; your GPS reading will give you an altitude, but relative to what? A geoid (from Greek ge, the earth) defines nominal sea-level, i.e. height 0, the base level in terms of altitude at any given x/y coordinate. A datum (Latin for 'that which is given', the singular of 'data'), was originally the set of reference points ('data') against which points on the earth's surface were measured, both horizontally and vertically. The term is now used in various ways, but is defined by the Ordnance Survey as "a set of parameters which defines the coordinate system and states its position with respect to the Earth's surface". The term terrestrial reference system (TRS) is also used for this. Combine these elements together - a coordinate system, a datum, and a geoid - and you have a geodesic or geodetic system (from Greek ge + daisis, division).
Reference Systems
Historically, different countries used locally based datums and reference systems, so they varied widely. This is gradually changing as more countries adopt worldwide standards, and the best-known geodetic system nowadays is WGS84, the World Geodetic System defined in 1984. This is the system used by GPS, i.e. geocentric, with Greenwich as central meridian. It is basically a modified form of GRS80, the Geodetic Reference System of 1980.
Well-known reference datums are NAD83, the N American datum of 1983, and ED50, European Datum of 1950. ETRS89, the European TRS of 1989, is WGS84 fixed on Europe, i.e. as the Eurasian plate moves, ETRS diverges from WGS. (It's another complication that tectonic plate movements mean that relative positions on the earth's surface are continually changing, both horizontally and vertically, so models have to be revised periodically.)
Coordinate Systems
Your GPS device will give you geographic coordinates, angular measurements in degrees of latitude and longitude (and altitude). These are segments of the ellipsoid. Latitude is based on the equator (0°), with north being positive and south being negative. There is no such obvious line for longitude, but in 1884 the Greenwich meridian was adopted as the prime meridian (0°), with east being positive and west being negative.
The WGS84/GRS80 ellipsoid uses an equatorial radius of 6,378,137m. Multiplying by 2π gives you a circumference, i.e. equatorial length, of 40,075,016.686m, 20,037,508.343m for each hemisphere, or 111,319.491m per degree.
That's true of the equator, but the distance between lines of longitude will change as you move away from the equator. As discussed on the Where Am I page, on a map you need a scale model of the earth's surface, where all distances are at the same scale. Your coordinates are then standard x/y planar coordinates, which nowadays are mainly in metres.
A complication is that geographic coordinates are generally given in the order latitude/longitude, i.e. y/x, whereas planar coordinates are normally x/y.
External Resources
There is a good tutorial on geodetic and coordinate systems on the Ordnance Survey site (though once you've digested that lot, you may wish you hadn't asked).