Trophies are one of the rewards that you get for winning PvP battles.
At the end of each PvP Season, all Trophies you have earned this season will earn you Gold and War Coins at a 1:1 ratio, and 1 Diamond for each 50 Trophies.
Furthermore, the number of Trophies is used in two different Leaderboards:
Only the Top 100 are publically displayed in any leaderboard. You can always view your own rank even if you are not part of the Top 100.
The amount of rewarded Trophies will increase with more difficult PvP battles. The difficulty of any PvP battle is determined by the difference between your Army Power of your top 5 regiments and the Army Power of your selected opponent. The difficulty is displayed in the form of 1 to 4 skulls. The more skulls, the higher the difficulty and rewards!