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.
Furthermore, if the point is attached to the Earth (e.g., a mountain peak), then altitude (height above sea level) is called elevation.
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.
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