How to trigger quest:
Ok, to be honest this isn't even a quest in the game. That's right, not even a 'miscellaneous quest'. However, I've been emailed and asked about these Dragon Priest Masks enough that it's about time I added this "quest" in.
Basically, throughout the game you will only fight a limited number of Dragon Priests. Well, a limited number of NAMED Dragon Priests anyways. Eight, to be exact. Each of these Dragon Priests drops a special mask that is often enchanted with kick-ass enchantments. Collecting them all is essential to this quest. Once you have them all, you must take them to Labyrinthian where you can find a building in the center of an area with an unusual statue displaying NINE busts of Dragon Priests. Here you will also find a skeleton along with an intriguing note and a WOODEN MASK. You can equip the wooden mask here, in this room with the statue, to be TRANSPORTED BACK IN TIME! The idea here is to have all eight of those Dragon Priest Masks and go back in time and put them on the statue.
So, without further ado, here's where you can find each of the eight Dragon Priest masks. Note that several of them fall into particular quests, so you may as well do the quest associated with the Dragon Priest fight.
You will fight the Dragon Priest Hevnoraak during the quest "Evil in Waiting", one of the game's many side quests. Simply visit the ruins of Valthume (Southeast of Markarth, southwest of Rorikstead, and west of Cradle Stone Tower) and talk to the ghost you find to start the quest that will wipe out Hevnoraak.
Krosis can be found near "Shearpoint", a mountain tip located to the west of Windhelm. He guards a Dragon Wall (the Dragon Wall for "Throw Voice") up there, so head up there and get yourself a Dragon Priest mask and the words for Throw Voice already!
Morokei is tied directly into the College of Winterhold questline. You will fight him in the quest "The Staff of Magnus", as he is the one carrying the staff. It'll take a little while to get to him if you haven't done the College of Winterhold questline, but I'd recommend just doing that questline to get to Morokei.
Nahkriin is tied in directly to the Main Quest. You battle him near the end of the quest "The World Eater's Eyrie". It's a great battle, but going through the main quest is the ONLY way you can reach him, as the zone he is in is only accessed that way. So, this will be another mask you'll have to devote some time to getting if you haven't done any of the main questline yet.
You can fight Otar while doing the side quest "Unlock the Sarcophagus in Ragnvald". Ragnvald is a set of ruins located northeast of Markarth, so head on over there and you'll have to locate skulls to make Otar show up, but I cover that quest in much greater detail in the side quest section, so check it out if you need help.
Rahgot is tied to the quest "Siege on the Dragon Cult", which you obtain from an Imperial NPC upon visiting the ruins of Forelhost, which are located to the south of Riften. Once again, I go in much greater detail in the side quest itself, but you NEED to activate this side quest to get INTO the ruins, so you may as well go along with it.
You will face Vokun when you go and do the side quest "A Scroll for Anska". You get this quest by visiting the High Gate Ruins for the first time (which are located halfway between Solitude and Dawnstar, north of Ustengrav). You'll find an NPC there named Anska that you'll get the quest from, so with her help head in and defeat Vokun.
Volsung is located at Volskyge Peak, which is northwest of Dragon Bridge and west of the Statue to Meridia. You CAN get up there by climbing the mountain, but to do it "properly" you'll need to go through the ruins at the bottom of the mountain. It's pretty straightforward, but in the ruins you'll encounter a puzzle. The solution to it (if you DO go through the ruins) is SNAKE, BEAR, FOX, WOLF. At the top of the peak you'll discover a word wall where you'll learn a word for the shout "Whirlwind Sprint". Once you learn the word, Volsung will emerge from a nearby crypt.
Once you have all eight of those masks (in your inventory!), head over to Labyrinthian and find that room. Put on the wooden mask and teleport back in time, then put the eight Dragon Priest masks on the bust. Once you have put all eight of them on, the NINTH bust will show itself and allow you to take a NEW mask:
When health is low, has a chance to heal wearer and damage nearby enemies. You can now grab ALL NINE of the masks and return back to Skyrim. That is the end of this "quest". Use all of the masks as you see fit.