In the United States, "yes" and "no" are the common, ordinary, basic words that we all use every day.
"Aye" and "nay" are old-fashioned, unusual, and used only in few specific situations.
The only place where I've heard it used is in a formal meeting where a "voice vote" is being taken. The custom is for the chair to say "All those in favor, say 'aye!'" And a lot of people say "Aye!" He then says "All those opposed say 'nay!'" And people say "Nay!" The chair looks, listens, and judges, and then announces "The ayes have it!" or "The nays have it!"
In the U.S. Navy, and probably other English-speaking navies, I think it is customary to say "Aye, aye, sir!" in response to an order. It means more than "yes." It means "I acknowledge that this is an order, I understand what I'm supposed to do, and I will do it."