The quest interface takes a bit of getting used to in The Secret World. You can click on the main icon to get a drop down of all your quests. Clicking on another quest will make it the active one. Clicking again will bring a pop-up.
Types of quests we’ve discovered so far:
Story – blue icon, primary game narrative (you only have one at a time)
Group/Dungeon – purple icon, dungeons (you only have one at a time)
Action – red/brown icon, combat-focused (you can have up to three side quests)
Investigation – light green icon, factional history/challenging puzzles (you can have up to three side quests)
Other side quests such as kill & explore (you can have up to three side quests)
You will only ever have one story arc quest (the top one). You can also only ever have one of the dungeon quests at a time, as well as the action quests. You can have three side quests at any given time. If you already have a quest of one type and pick up a new quest of the same type, it will go on hold and you can resume it from your log (L) at any time.