There are two types of rewards (in addition to gold and XP) that you can receive from winning a PvP battle: Banners and Trophies.
You receive banners when you defeat an opponent in a PvP Event.
Banners mark your personal progress in an Event. When you accumulate enough banners, you will advance on your reward ladder until you reach the top reward. After that, you will still gain banners, but won't earn additional rewards from them. You can tell if you're in an event by looking at the bottom-right of the opponent selection screen. If you are in the event, the rewards icon will be the Banner shown above.
Banners aren't permanent. Once the event that you earned them from ends, they disappear. When a new event starts, you will begin with zero banners again.
Note that in some events, you can lose Banners if you battle another player and lose. You do not lose Banners if you are defeated in a PvP battle outside of an event.
You receive Trophies when you defeat an opponent in a PvP battle outside of a PvP Event.
When you hit the Battle button, instead of entering an event, you go to select another player for a direct Player vs. Player fight. The opponent selection screen is similar to the one you get when you're in an event. However, outside of an event, the rewards of the battle, which you can see in the bottom-right of the opponent selection screen, will have the above Trophy icon as a reward (not the Banner icon).
Trophies don't win you any rewards, but they are used to gain the different levels of the Bloodthirsty Achievement
Unlike Banners, Trophies stay with you. But if you lose a fight with another player, you will lose trophies, so the number of them that you have can go up and down depending on how you're doing in battles.
You don't lose Trophies if you are defeated by an opponent in an event.