Each spec has a unique weapon and unique quests tied to it.
Inspiration for Artifacts--fighting ultimate enemy needs ultimate weapons.
Examples of unique quests--Ret Paladins return to Broken Shore and fallen heroes to retrieve Ashbringer, Protection Warriors discover legend of Vry'kul King, Death Knights retrieve shards of Frostmourne from base of Icecrown.
These weapons are with you through Legion--you level up with them. Gain Artifact Power as you level, which is used to unlock traits.
Artifact weapons have a special UI--you eventually unlock all traits but you can determine the order. Some traits are more powerful than others.
You can also customize weapon--both design and tints. You can unlock them through a variety of ways--some will be via leveling up your weapon, but others will be via raids, PvP, or more completionist activities. Everyone shouldn't look identical, the weapon's appearance should reflect what you like to do in game.
When your weapon gains a certain amount of power, you unlock a "master" appearance of that weapon.
Ret Paladins--Ashbringer. Fiery, Shadow, Lightning variants.
Monks--Sheilun, Staff of the Mists. Other variations inspired by animal gods of Pandaria and the Sha.
Fire Mage--Felo'melorn. Weapon Kael'thas used to fight Arthas in Icecrown Citadel. It's been lost, until now.
Survival Hunter--The Eagle Spear. Highmountain Tauren lore background--you defeat the beast that defeated the weapon's last owner. Other variations inspired by wolf, serpent, bear. Spear implies Survival Hunters are now melee.
Frost Death Knight--Ice Bringer and Soul Reaper. Forged from remains of Frostmourne. These are more traditional and based around Death Knight fantasy--skulls, frost, more skulls, different skulls.
Enhancement Shaman--Doomhammer. Thrall's Weapon. Variants are Fel Version, Mag'hari, Magical Crystal.
Feral Druid--Fangs of the First Nightsaber. Changes Druid Form!
The Legion website has provided a more in-depth look at several of the weapons:
Ashbringer: It is said that this sword, forged by Magni Bronzebeard, bestows the glory of the Light upon its wielder, empowering them to render enemies into ash. Paladin weapon.
Maw of the Damned: For eons, the demon known as Gorelix the Fleshripper used this massive axe to steal the life force of his enemies and replenish his own. The Maw will bleed dry anything it touches--eating is all it knows. Death Knight weapon.
Ebonchill, Greatstaff of Alodi: The staff of the first Guardian of Tirisfal exudes unnatural iciness, allowing its wielder to keep a cold, sharp, and collected mind even in face of overwhelming odds.
Sheilun, Staff of the Mists: This legendary staff was used by the last emperor of Pandaria to shroud his homeland in mist, and it resonates with his everlasting legacy of wisdom and perseverance.
Thas'dorah, Legacy of the Windrunners: Legend has it that his elven family heirloom, once wielded by an infamous Ranger-General--can turn a mediocre archer into a master marksman..and make a master truly peerless.