Quests are interactive stories in which one person posts pictures and events, and the other posts decide how to behave. This produces a collaborative story.
Quests are run on tgchan's /quest/ board, or sometimes, 4chan's
Completed quests are gathered in the Completed Quests category; this is for quests with multiple threads and moderate- to long- running times that have reached a conclusion. Shorter quests are often referred to as "one-shot" quests, and are set aside from the rest of the quests in their own category.
There also exists a formal page for assembling a list of all the major quests, but it tends to be less up to date than the category indexes. For that page, see: Running Quests
Quest quest quest.