Synonyms: conquer, triumph, win
Prevail means to successfully persuade someone of something. If you were a Presidential advisor and you convinced him to make a National Day of Pet Appreciation, then you prevailed upon him to recognize pets.
Prevail can be used for different kinds of success. If you prevail upon someone, you have influence on them. If you prevail over someone, you win. Say you’re a big fan of a losing team. You might cheer, “Despite all odds, we will prevail!” To use prevail, there has to be some kind of contest or conflict. That could be internal, like when you have a pile of homework to do, but your desire to play video games prevails over your discipline.