You use your ability cards to work on quests in LoN. On almost all ability cards, there will be a number with what looks like a starburst around it. Usually it's a 1 or 2, sometimes a 3. Then, on each quest, you'll notice a circle on top and bottom of the quest card. The bottom circle is the one you have to fill up in order to complete the quest (the top is for the computer or other player). So if you play an ability that has a 2 on it, you should notice that your circle then has 2 dots in it. For the starting quests, 2 dots is enough to "complete" the quest. The next quest will require 4, then 5 and finally 6 dots to complete.
Make sense? haha.