In the arena you are faced with 4 opponents to pick from. from this you have 5 attempts to defeat one. You can refresh your attempts through using gems. Each time you defeat an opponent you will take the place of the person and your opponents list will refresh. Each win rewards you with a small amount of glory points. It is key to observe each deck displayed and figure if your deck can take on the opposing players.
Gem & Conquer Edit
Every time you move up a rank you earn gems. in theory you can win more gems than you can spend on Attempts, but you can only purchase 5 attempts a day. This falls off certainly later when you cannot defeat the opponents you are faced with also you don't win gems for going over old ranks. So if you are rank 2555 and you drop 450 ranks you are at 2105, when you go up a rank and hit 2500 you do not win any gems as you previously did for going past or at that rank. You only earn gems from placing above your highest rank achieved.
How Many Arena's are there? Edit
There is current evidence which shows that there is actually more than one Arena (Amount is currently Unknown). Each Arena seems to only have a max of 10,000 ranks. Therefore there is actually for certain no one supreme Champion in the Deck Hero's. Only global stats or ranks are taken from Trails Rankings and Event Rankings.
Through battle log you can see how you lost defensive Arena Matches in your absence and is a good way to determine upgrades and optimizations for your deck and identify your decks weaknesses.