Charm quests are the tasks associated with charms recieved either as drops or as rewards from glowing obelisks. There is an achievement for completing 20 charm quests.
There are 3 levels of charms, level one being white, 2 purple, and 3 red. Red charms, aka epic charms, are the best available and have the most beneficial bonuses that seem to occur more often than the other charms.
Completing charms involves doing the task associated with it as well as inserting the necessary runes or other charms into them.