The third gameplay mode Dissidia offers isn't even available until story mode is complete. The Duel Colosseum is basically an extension of it, whose primary purpose is to provide access to materials needed to create high-level equipment. This is Dissidia's end-game.
And would you believe that it's a card game? Story mode was a board game, so why not, right? When you enter the Duel Colosseum, you draw a hand of three cards, which come in several different types. The first type is the most obvious: battle cards. These cards can be used to initiate a battle against the characters depicted on them with predetermined conditions. Successfully beating the opponent awards medals, while a loss costs medals. Losing a battle also decreases your hand size by one, and if it gets down to nothing, the only thing you'll be able to do is quit and restart.
Medals are the bread and butter of the Duel Colosseum, and can be spent to purchase the items depicted on treasure cards. Each treasure card contains an item and costs a certain number of medals to use. Using it immediately puts the item in your inventory where it can be used to craft new equipment.
The third main type of card is the job card. Each depicts a different Final Fantasy job class, each of which grants a different useful effect. Knight cards increase your hand size, letting you hold more cards, while thief cards increase the likelihood of drawing treasure cards. Only three job cards can be in effect at any given time. There are also status ailment cards, like toad, which contain various negative effects. The effects of these cards only occur if you use them, which isn't likely to happen unless you only have one card left in your hand.
Every time you use a card, your entire hand is discarded and a new one is drawn. There is one exception to this rule, however, and that's boss cards. These special battle cards contain more powerful enemies with stronger AI and equipment, and until you fight them, they'll remain in your hand, wasting valuable space.
Duel Colosseum continues until you decide to stop or you no longer have a hand of cards to play. There's no victory, you simply keep playing until you've had enough or you lose too much, at which point any extra medals you've accumulated will be converted to PP, which can be used to unlock characters, icons, and various game bonuses in the PP Store.