All of a quest's objectives must be finished before you can complete a quest. These objectives can be tasks such as killing monsters, gathering items, or visiting certain areas. After a quest's objectives are finished you can complete the quest by talking to the NPC or by interacting with the object that is indicated by the quest. They will have a green checkmark above them.
Completing a quest can reward you with many things. You will always gain experience, and you will almost always gain money. Additionally, you will sometimes be rewarded with reputation with a certain faction, or with a piece of armor or a weapon.
Quests that you no longer want to complete can be abandoned by clicking "Abandon Quest" at the bottom right-hand corner of the Quest Log. You can also share any quest in your Quest Log with other members of your party by clicking "Share Quest" at the bottom right-hand corner of the Quest Log.