Let's give a good example of a quest, say Quest for the Gemblades. This is its description:
Name: Quest for the Gemblades
Casting Cost: (1)(Forest)
Abilities: Whenever a Creature you control deals damage to an opponent, you may put a quest counter on Quest for the Gemblades. Remove a quest counter from Quest for the Gemblades and sacrifice it: Put four +1/+1 counters on target creature.
This quest requires one Mana of any color and one Green mana to cast. It is in Worldwake edition and is a green card. It is also an Enchantment. The abilities are also good, since they say that whenever a Creature you control deals damage to an opponent, you may put a quest Counter on Quest for the Gemblades. Then, one you "finish" the quest by getting one quest counter, you can remove it and sacrifice Quest for the Gemblades since you've finished the quest. But for finishing the quest, you can put four +1/+1 counters on target creature! That's pretty good. There are lots of good quests out there.
Quests are not the only cards that give quests counters. Cards called "Ascensions" can do this too. For example, take Beastmaster Ascension. Here it is:
Name: Beastmaster Ascension
Casting Cost: (2)(Forest)
Abilities: Whenever a creature you control attacks, you may put a quest counter on Beastmaster Ascension. As long as Beastmaster Ascension has seven or more quest counters on it, creatures you control get 5/5.
This card is VERY good! It is a Rare, which proves its value. It's three mana to cast, and it is Zendikar edition. And whenever a creature you control attacks, you can put a quest counter on it! And when it has seven or more, creatures you control get +5/+5! That is, you don't have to sacrifice Beastmaster Ascension in order to activate the ability. Ascensions commonly have abilities that are different from quests in the way that you can't use them with quests; it's too confusing.