The scale height is the perpendicular distance over which a particular physical variable drops by a factor of e (~2.71828). It is heavily dependent on the type of material/object you are observing.

In a spiral galaxy, it is the height above the galactic disk at which the density of a particular constituent of the disk has declined by e. This is different for different types of object (e.g. the scale height of stars is different to the scale height of HI gas in the galaxy), and generally increases with the age of the object.

In an atmosphere (be it planetary or stellar), the scale height is the vertical distance over which the atmospheric pressure drops by a factor of e.