A creature is modified if it has a counter on it, is equipped, or is enchanted by an Aura controlled by that creature's controller. Modified decks combine those common upgrade methods under a single mechanical umbrella.
Kosei, Penitent Warlord wants several different kinds of modifications at once, while Invigorating Hot Spring supplies +1/+1 counters and haste. Equipment and Aura decks can therefore support modified payoffs without changing their underlying strategy.