The orthometric height of a point is the distance H along a plumb line from the point to a reference height. When the reference height is a geoid model, orthometric height is for practical purposes "height above sea level".
In the US, the current NAVD88 datum is tied to a defined elevation at one point rather than to any location's exact mean sea level. Orthometric heights are usually used in the US for engineering work, although dynamic height may be chosen for large-scale hydrological purposes. Heights for measured points are shown on National Geodetic Survey data sheets, data that was gathered over many decades by precise spirit leveling over thousands of miles.
Alternatives to orthometric height include dynamic height and normal height, and various countries may choose to operate with those definitions instead of orthometric. They may also adopt slightly different but similar definitions for their reference surface.
Since gravity is not constant over large areas the orthometric height of a level surface other than the reference surface is not constant, and orthometric heights need to be corrected for that effect. For example, gravity is 0.1% stronger in the northern United States than in the southern, so a level surface that has an orthometric height of 1000 meters in Montana will be 1001 meters high in Texas.
Practical applications must use a model rather than measurements to calculate the change in gravitational potential versus depth in the earth, since the geoid is below most of the land surface (e.g., the Helmert Orthometric heights  of NAVD88).
GPS measurements give earth-centered coordinates, usually displayed as height above the reference ellipsoid, which cannot be related accurately to orthometric height above the geoid without accurate gravity data for that location. In the US, NGS has undertaken the GRAV-D ten-year program to obtain such data with a goal of releasing a new definition in 2022.