Scale Height is a measure of the decrease of something that falls off exponentially, specifically, the distance over which it falls by a factor of e (~ 2.718). Examples can include:
For example, a planet's atmosphere might fall by a factor of e every 10km of height. Despite the word "height", it might be applied to dimensions not thought of as vertical.
Equation forms associated with scale height:
A = A0e-z/H A z ln —— = - —— A0 H dA —— = -HA dz
(The first two are forms of the solution to the differential equation.)