Abandon means that you are able to remove the quest from your journal listing of quests. With the exception of Campaign quests, nearly all quests can be abandoned by players. If the quest began from an item, deleting the item will also abandon that quest and the quest cannot be reacquire without getting another item. Sometimes you are given item(s) during a quest. These items will not always be removed from inventory when abandoning the quest. When you abandon a quest, it is a good idea to check if an item related to that quest still exists in inventory.
To abandon a quest, select the quest in your journal. At the bottom left of the screen there will be an Abandon button. A confirmation window will pop up to prevent accidentally abandoning quests. To get an abandoned quest again, speak to the NPC that offered it or find the item that began the quest.