Quests in Terrasphere are exactly like in video games, they provide you with objectives to follow, giving you a clear direction to approach in your thread. Since most quests provide some sort of dice rolling or mechanical challenges, you may fail a quest.
Typical quests are not unique, multiple groups or characters can embark on the same quest. If you find a quest you might like doing with others, post in the LFG forum to find a group.
Any quest that is tagged as "story" or "world" quests are unique and time-limited quests that are tied to the main story or the current event. These quests often offer small rewards for completion and affect the story or world in some way.
For more information on other roleplay content, check out Roleplaying (Wiki). If you would like to write quests, see this thread.