It's used as shorthand for how loud an amp will go before you get overdrive. Some amp's don't have much - like a Fender Champ. People using with a low wattage amp often want more headroom to be heard over a drummer when playing live (i.e. they crank the amp, but it distorts leaving them without a clean tone).
I had a Diezel that could do 160W of clean. Now that's headroom
As hear and play states, headroom has more to do with the circuit design and component values including the transformers. Other elements to a certain degree would be the tubes and the speakers as to how clean an amp can stay after the circuit has been designed for clean headroom.