Artifact weapons are going to be one of the key features of Legion. I suspect that Relics will simply be the new weapon drops and the Artifact Trees will essentially be like the Draenor perks, only that we get to choose the order of them and there will be more of them. In other words, it will mostly be a lore/cosmetic thing.
But it's still pretty exciting (even if I am worried about Frost DKs once we're forced to dual-wield.)
There will be 36 sets of Artifact weapons - one for each individual spec. Given that there won't be weapon drops as usual in Legion, I think it's safe to assume that any one-handed weapons will be paired with an off-hand weapon, frill, or shield. Presumably these will be cosmetically customizable as well (though I don't know if you'll be able to do them separately if you want an eclectic look.)
So far, only a handful of Artifacts have been revealed or talked about, but here's what we've got:
Retribution Paladin - Confirmed
Using a shadowy crystal dropped by an Orc Warlock that was purified by a group of Paladins, Magni Bronzebeard forged the Ashbringer to serve as the greatest weapon ever used to fight the Scourge. Tirion Fordring ultimately sundered Frostmourne with the Ashbringer, releasing the souls trapped within and allowing the heroes of Azeroth to slay Arthas, the Lich King.
Doomhammer and Shadowy Echo of Doomhammer (name unconfirmed on the latter)
One-handed Mace (x2)
Enhancement Shamans - Confirmed
A great hammer imbued with the power of the primal elements of Draenor, the Doomhammer was passed down through the family line that bore its name, and was the weapon that slew Anduin Lothar. When Orgrim Doomhammer died, he passed his ancestral weapon along with the mantle of Warchief to a young Thrall.
Icebringer and Soulrender
One-handed Sword (x2)
Frost Death Knight - Confirmed
The cursed Runeblade Frostmourne was created by the original Lich King, Ner'zhul, to steal the soul of Arthas Menethil, who would eventually use the sword to shatter the Frozen Throne and merge with the Lich King. However, when he attempted to raise the heroes he had just killed to serve as his greatest champions at the climax of the Icecrown Citadel raid, Arthas' weapon was shattered, releasing his father's soul, who resurrected the heroes and immobilized him while they finally brought him to justice. As a Frost Death Knight, you'll track down the shards of Frostmourne and reforge them into a pair of rune swords to use their dark power to fight the Legion.
Sheilun, Staff of the Mists
Mistweaver Monks - Confirmed
Sheilun was the staff of Pandaria's Last Emperor, Shaohao, who hid his land away during the War of the Ancients with a magical mist as he became one with the land.
Felo'melorn and as-yet-unknown off-hand frill
One-handed Sword and Frill
Fire Mages - Confirmed
Kael'thas Sunstrider took his Blood Elves to battle the Scourge alongside Illidan and Vashj. When the battle was lost and Illidan's forces had to retreat back to Outland, the Prince of Quel'thalas lost his sword in the frozen wastes of Icecrown.
Fangs of the First Nightsaber
Unknown Weapon Type
Feral Druids - Confirmed
We're beginning to get into weapons we don't really know the lore for. It's unclear whether this will be a staff/polearm or if Feral will be given the ability to dual-wield and thus these could be fist weapons.
The Eagle Spear
Survival Hunters - Confirmed
Likely wielded by Huln Highmountain, leader of the Highmountain Tauren, who aided in the effort against the Burning Legion during the War of the Ancients.
Maw of the Damned
Blood Death Knights - Strongly Suggested
This axe was wielded by a demon named Gorelix, who used it to draw the lifeforce off of his enemies - so it seems likely that it's for Blood Death Knights.
Thas'dorah, Legacy of the Windrunners
Marksmanship Hunters - Strongly Suggested
This bow is presumably the one used by at one of the Windrunner sisters, though which one is up in the air, as all three could potentially play a role in the expansion. I'd only say probably not Sylvanas as we know she'll be active during the expansion and probably has a darker-themed weapon. It's also very likely for Marksmanship due to the use of the word "Marksman" in the description on the website.
Ebonchil, Greatstaff of Alodi
Frost Mages - Strongly Suggested
This is the staff of the first Guardian of Tirisfal, the super-powered mages created to protect the world from the demons of the Burning Legion. Given the name and the description making a lot of puns about keeping cool, this really seems like it will be for Frost Mages.
Icebreaker and as-yet unknown shield
Sword and Shield
Protection Warriors - Confirmed
An ancient Vrykul king had a sword and shield made using a scale from Neltharion before he was corrupted by the Old Gods. The shield will literally be one of Deathwing's uncorrupted scales. Vrykuls and Dragons - somehow they fit Warriors really well.
Arcane Mages or Some Priest Spec - Possibility
There's a lot of speculation on this one. In the preview video for Legion, the character wielding this has mage gear on, and this has the purple arcane look to it (also we already know the other two mage weapons.) Still, some have suggested, especially given the draenei crystal look of the staff, that it could be for Priests. Then again, Draenei have Mages (and before the Legion came, their people were famous for their magic.) I'd say it's likely the Arcane weapon.
"Fang" (likely a placeholder name, or needs a subtitle)
Assassination or Subtlety Rogues - Possibility
Very little known about this one - it looks like there's a little spherical well at the hilt, which could be for poison, and thus suggest Assassination. Still next to nothing about it.
Two-Handed Sword, Probably
Unholy Death Knights - Possibility
We have an image of a sword with a skull and green runes on it. This seems to strongly suggest it's for Unholy Death Knights, but it's rather thick and could thus be a one-handed sword, possibly for some Warlock spec. Still, I think the runes and skull make it more likely as the Unholy weapon, and the green color would fit just as well with Unholy as it would with Warlocks.
As-Yet-Unknown Paladin Mace
Protection or Holy Paladins
This is clearly a Paladin weapon, as it has a theme that very clearly lines up with the Judgment set (tier 2, found primarily in Blackwing Lair,) one of the most iconic Paladin sets. This might be an alternate skin, though, so what the weapon looks like baseline or what its backstory is remains unknown. And while I think Protection is more likely to have a sword, with Holy getting a mace, it's certainly possible that it could be the tanking weapon. (Just please don't give us an axe. Axes feel so wrong for Paladins.)
Beast Mastery Hunters - Possibility
With Marksmanship getting a Bow, and Survival getting a Polearm, it seems likely that Beast Mastery will get either a Crossbow or Gun. We don't have any confirmation that another class will be using ranged weapons, and the image of the gun seems to have tusks, so this seems very likely to be the Beast Mastery weapon.
So, all together that's 17 (if I counted right,) which is just under half of the Artifacts coming in Legion. Whoo-boy. I imagine a lot of them are either still being decided or, more likely, still being worked on and not in a state to show off yet. We also know that some of the Glaive models we saw in the Demon Hunter previews will be the artifacts for Havoc or Vengeance.
I believe their policy is going to be to give us weapons that we have not yet been able to equip in-game (though the no-longer attainable Corrupted Ashbringer might beg to differ,) and as such I don't expect to see existing legendary weapons to come back as artifacts. So Demon Hunters won't get the Blades of Azzinoth (at least not as their artifacts - I'd love it if Illidan was given some special dialogue for Demon Hunters who went back to run Black Temple,) but there will be plenty of demons to kill for their weapons. Of course, that could be pretty restrictive - Gorehowl and Ashkandi would both be great Arms Warrior options if they didn't already exist in game (multiple versions, sort of, of both.)