What does the word "hard" in this context mean? Something aggressive or determined, I suppose?

Rebecca

Yes, your guess is correct! A "hard stare" is a very steady, focused stare with a lot of attention or determination. For example, if you wanted to make a facial expression that was challenging someone, you would give them a "hard stare." Ex: The athlete gave his competitor a hard stare. Ex: I gave her a hard stare to prove that I wouldn't be intimidated.