A "stay" is one or more continuous nights at the same property. (And note, for example, if you make three separate reservations for three successive nights at the same property it's still going to only going to count as one "stay." If you alternate different nights between two properties over,say, four nights it'll count as four stays)
A "night" is an, er, night!
Edit: Kiwi beat me.
And...
Originally Posted by Kiwi Flyer
...In some cities you could spend each night in a different hotel to quickly rack up stays.
My first Diamond challenge which was x stays or y nights, I alternated between four different hotels in SAN over the course of six nights in order to get six nights AND stays.