What is the measurement of height above Earth's surface?

The height above the ground is called altitude. The Word is sometimes used to mean elevation, which is the height above mean sea level.

In aviation, altitude is usually expressed in feet ASL (above sea level), but can also be stated as feet AGL (above ground level) for purposes of avoiding obstructions. Properly set barometers can determine height above MSL, while radar altimeters are used to measure AGL.