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Your quest log keeps track of the current set of missions that your character has accepted. The default key for accessing the Quest Log is L, or you can click the chalice button on your menu bar.
The quest log will list each of your active quests, broken down by zone they are associated with. Within a zone, they are usually listed by difficulty, and then alphabetically. Normally a quest's name will be in yellow to indicate average difficulty. Orange and red are more difficult for a character at your current level, while green should be easy and gray is trivial. Gray quests offer almost no experience as a reward for finishing them.
By clicking on each quest, you will find information about the quest, including the objectives, the progress you have made, and what the questgiver said to you about the quest. At the bottom of the window are options for sharing the quest with nearby group members and abandoning the quest altogether. Since you can have a maximum of 25 active quests, it is often important to abandon old quests. The quest log displays the number of current quests at the top.
When you make progress toward a quest, in addition to a very brief notice in the center of your screen, a small reminder will appear down the side of your screen to mark your overall progress with the quest. This will usually fade after about 5 minutes of making no additional progress. If you want to display that quest's objectives permanently or hide the quest's objectives, shift-click on its name in the quest log.