When a player takes on a quest, it will remain in their quest log until it is completed or otherwise removed. Because there is a limit on the number of quests a player can have active, this will mean that occasionally the player will want to drop one or more of their quests entirely. To do this:
If a player abandons a quest, they can always reinitiate the quest later from the questgiver. If the quest was part of a chain, they will be in the same position they were before, meaning they need to go to the appropriate giver of that particular quest.
Also, when a player abandons a quest, the game may remove the Quest Items associated with it. If the player takes the quest once more, they will regain those items -- this is sometimes an important reason to abandon a quest. Items that are given at the start of the quest and consumed or used up can sometimes force a player to take the quest a second time to get the item back.
While only certain quests, particularly escorts, can be failed, if a player wishes to try the quest again, they typically need to abandon the original failure.