Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For an example of vertical extent, "This basketball player is 7 foot 1 inches in height." For an example of vertical position, "The height of an airplane in-flight is about 10,000 meters."
When the term is used to describe vertical position (of, e.g., an airplane) from sea level, height is more often called altitude.[1] Furthermore, if the point is attached to the Earth (e.g., a mountain peak), then altitude (height above sea level) is called elevation.[2]
In a two-dimensional Cartesian space, height is measured along the vertical axis (y) between a specific point and another that does not have the same y-value. If both points happen to have the same y-value, then their relative height is zero. In the case of three-dimensional space, height is measured along the vertical z axis, describing a distance from (or "above") the x-y plane.
Note that altitude usually refers to a height in the air (above sea level) and elevation refers to height on the surface [of the Earth] above (or below) sea level.