As the name suggests, ellipsoidal heights are measured in relation to a reference ellipsoid rather than sea level. For NZGD2000 this is the Geodetic Reference System 1980 (GRS80) ellipsoid.
Ellipsoidal heights do not relate to the local gravity field of the Earth nor mean sea level (MSL). This means that ellipsoidal heights may be non-zero at sea level and cause water to appear to run uphill. NZGD2000 ellipsoidal heights are approximately equal to MSL near Stewart Island/Rakiura and differ by around 35 metres in Northland.
NZGD2000 ellipsoidal heights can be related to NZVD2016 (and the 13 local vertical datums) by using the NZGeoid2016 geoid model. The conversion of heights between different systems is described on the height conversion pages.
Global navigation satellite systems (GNSS), such as GPS, measure ellipsoidal heights. This is because mathematical calculations on the ellipsoid are simpler to carry out than on the undulating mean sea surface. Some GPS receivers will output heights in relation to sea level, however to derive these they will have applied a geoid model to transform the ellipsoidal heights.
The accuracy of ellipsoidal heights is classified by a series of Orders. The Order of a height is related to the accuracy of the observations used to derive it and the type of ground mark that is used to physically represent it. The Order classifications between vertical datums are not consistent.