Artifact weapons are going to be a core mechanic of Legion as an expansion. We're going to see our weapons leveling up with us, and while I don't have all the details, there does seem to be a bit more to explain the system now.
You'll upgrade your artifacts through two means - Relics and Artifact Power.
Artifact power will, I believe, be rewarded for various things - I believe that no matter what you do in-game, you'll be able to acquire artifact power, though it's possible that certain methods will yield greater results.
Artifact power is used to fill out the "talent tree" for the artifact. You'll unlock a weapon-specific skill right after you get the weapon, and then you'll be able to fill out various bonuses that increase the damage of certain abilities or function somewhat the old pre-Mists talents. Each talent you fully upgrade will give you access to the ones branching off of it, and eventually you'll be able to unlock more interesting ones signified with gold dragon borders similar to an elite enemy.
I don't know if this artifact power will all be from a shared pool between your artifacts or if you'll need to gather it separately, but they do say that you should be able to have your whole talent tree filled out for your mainspec weapon after about three months of moderate play.
Relics will be found as drops in dungeons, rewards from quests, and other standard "gear acquisition means." Each artifact will have two different relic slots, and these slots will have a corresponding element. So far I've seen Arcane, Holy, Water, Frost, Fire, Blood, Fel, Shadow, Life, Wind, and Iron. So, for example, the Scepter of Azshara, the Restoration Shaman weapon, has a Life relic slot and a Water relic slot, whereas the Silver Hand, for Holy Paladins, has one Life slot and one Holy slot. Thus, Shaman and Paladin healers might both roll on a Life relic that drops off a boss, but other specs will be competing for the Holy or Water relics.
The relics appear to boost the weapon by a set item level (right now, they all only say +10 or +20, but I suspect that we'll just see bigger and bigger boosts as we get to higher levels and into endgame tiers,) and they also enhance one of the talents for the artifact. Right now the way that it seems to work is that you might pick up a relic that will boost one particular talent - such as "gives your Frozen Orb an additional 10% chance to crit," with three ranks, climbing up to 30%. You might then get a relic that raises that by a rank, so you'd be able to essentially have 4/3 ranks, with a 40% increased crit chance.
That should make this aspect of gearing a lot more interesting - sure, you might want the raw power of an item-level upgrade, but you might have to then take a relic that buffs some less exciting talent. It's a lot more complex than even balancing secondary stats.
It also means a lot more interesting gear competition, as I believe only a few specs overlap with the same two relic types (if any do.)
This system looks, at least at this point, really cool (obviously I'll have to get it in my hands before I can make a solid judgment on it.) I do really wonder what's going to happen when we give up our artifacts when we move on from Legion in a couple years. Things are definitely going to be nuts during Legion though, given that every single spec is going to have really unusual abilities granted to them by these fantastic weapons.