Later this system was formalized with a more exact definition. Also, the brightness scale was extended to negative magnitudes (like Venus, around -4, or the Sun, around -27), to stars or other objects weaker than magnitude 6 (which can be observed only in telescopes), and to fractional magnitudes.
Specifically, if two objects have a difference of 5 in magnitude, the brighter object is 100 times brighter than the weaker one; as a result, 1 magnitude difference is a factor of about 2.5 (fifth root of 100, to be precise).
The above refers to the apparent magnitude, i.e., as seen from Earth; an absolute magnitude can also be defined, i.e., how bright would a star or galaxy seem if it were at a standard distance? - The standard distance varies, depending on whether you want to compare planets, stars, etc. For stars, a distance of 10 parsecs is used as a standard.