You can play Conquest mode using Conquest tickets which can be obtained through purchase or by burning Silver Cards. Once in Conquest, you play until you lose or until you win three (3) matches. Based on how many wins you achieve during a Conquest, you get the following rewards:
0 win: no tradable rewards, only XP
1 win: 1 tradable Silver card + XP
2 wins: 2 tradable Silver cards + XP
3 wins: 1 tradable Gold card + 1 tradable Silver card + XP
Once you lose a match, your run will end and you will need to use another Conquest ticket to re-enter and start over.