The winner of a fight usually steals food, money, and gains tropheys for winning the battle. If you attack someone lower than your level and lose, then you will lose more tropheys than if the opponent had been a higher level. In other words, if you are level 20 and lose to a level 15, you will lose more tropheys than if you had lost to a level 25. Your number of trophies is effectively your pvp score, and gains you a spot on the leaderboards.
The Attack Log will show recent incoming attacks, whether you won or lost, and which resources were lost. You can also select your pvp team from this menu by clicking the CHANGE button on any monster you wish to swap out.
Building the Monster Arena will give you an additional attack in your stack. You start the game having only 2 attacks, and when you use one it starts to regenerate. The regeneration timer will start at 2 hours and count down until it is finished and you gain another attack. You have the option of speeding up the timer instantly by spending 5 gems.
You can search for an opponent by spending gold as you click the "SEARCH ENEMY" button. The amount you pay depends on your level. Once you reach level 20 the amount increases to 1000 per potential opponent. If you are attacked and lose you will automatically receive a 24 hour shield to protect from new attacks.
You also have the option to purchase shields with gems. A 1 day shield costs 1 gem; a 2 day shield costs 2 gems; a 7 day shield costs 5 gems. A shield can not be purchased twice without allowing it to expire. For example, if you purchase a 2 day shield then you must wait until it expires to purchase it again. However, at any time you could purchase a different shield and the number of hours will be added to your current shield duration.
NOTE: If you have attacked people they can take revenge at any time from their attack log. They can hit you even through your shield. I purchased a 7 day shield and it has 151 hours left, and I have received 3 attacks from people. The attack log says, _xxxxxxxxxx_ "TOOK REVENGE"