A quest is a task given you by NPCs. Quests usually have rewards and are a good way too receive experience beside leveling.
These kind of quests usually require you to get some items and bring them to the NPC.
They can include monster hunting.
Monster quests need you to kill monsters.
These may include: General monsters, Champion monsters, Giant monsters
"A man asks you to deliver a letter to a city guard."
That says pretty much all...
Take a couple of quests that are related to the same monster.
kill 50 Water Ghost/Water Ghost Slave
Find 20 Water ghost poison.