"The man is watching the boys outside."
Translation:A férfi kint figyeli a fiúkat.
For me, the english and the hungarian sentence say something different. In the hungarian one, the man seems to be outside and he is watching the children. In the english sentence, the man is rather looking through the window to the boys outside. I would prefer the translation: The man is outside watching the children. But I am not a native speaker.
No native speaker here but I think "néz" just means "to look at" whereas "figyel" means "to pay attention to, to keep an eye on".
Basically the former is just about perception, the latter also means (protective) surveilance or active listening.
So you "nézel" something observing it or studying it, but you "figyelsz" something or someone either to make sure it/he/she doesn't suffer any damage or run any risks or because you're receiving information from that source and you're actively listening.
For instance the English "listen!" or "look here!" you say for emphasis before something important would be "Figyelj" or "Figyelj ide"
I stand to be corrected if a native speaker has something to add or point out though.