Obtained cards can be equipped into the "Character Card Slot" in the character info window.As the character levels, the locked card slots will open in order And if you equip cards intothese slots,the card's power will be activated.
Level 01 - 1 slot opened.
Level 10 - 2 slot opened.
Level 20 - 3 slot opened.
Level 30 - 4 slot opened.
Level 40 - 5 slot opened.
Once equipped a card cannot be removed. However, you can replace the card with any new one.
In order to increase the rank of a card. The card must be from a monster higher than level 10. Visit any Card Merchant and you can upgrade a card. In order to do this, you must use 5 (five) different cards and place them in the wheel. (Order, AFAIK, is not important). You must also have purchased a Card Synthesis Stone from the Card Merchant in order to Combine a Card. You may also purchase from the Cash Shop, a Card Designated Synthesis, which will allow you to specify which card you would like to be returned to you upgraded, in the event of a success.
Now for some math. With a 50% success rate and a 20% opportunity to get the card you want returned to you, it's safe to say that you have about a 10% chance of obtaining the card you want when you combine cards. The Card Designated Synthesis removes that 20% chance and makes it a 100% chance and thus giving you the original 50% chance back. (Does not garauntee a success still)
The other alternative would be to simply place 5 cards that you want to be upgraded and well... one of them has a 50% chance and the rest will disappear. There are many ways to go about combining cards.