One of the key features of the Legion expansion is going to be the use of Artifact Weapons. The first major quest chain you get will be to get an artifact weapon (or set of weapons,) and you will continue using that weapon throughout the entire expansion.
Each specialization has its own artifact, and each artifact has its own story and background. Some of these will be familiar: for example, Retribution Paladins get the Ashbringer and Balance Druids get the Scythe of Elune. As many famous weapons as there are in Warcraft lore, there aren't really 36 famous weapons that would work for every single spec in the game, so many of these weapons are new inventions. That being said, each still comes with a backstory, and many of them have relationships with important figures of the past. The Shadow Priest artifact, for example, is a dagger and an off-hand book. The dagger is literally the claw from an Old God's (I believe N'zoth's) tentacle and was used as a ritual blade back in the era of the Black Empire - before the Pantheon came to Azeroth. Protection Warriors get a shield made from one of Deathwing's scales.
Artifacts will grow in power in two ways.
We'll start with the more complicated way.
Doing more or less anything in game (so questing, running dungeons, doing PvP, or even finding treasures or interested NPCs) will grant you artifact power. You can think of this a little like experience for your artifact weapon.
Each weapon has traits that can be unlocked at the cost of artifact power, with each subsequent trait costing more. Each weapon has one inherent ability (this "trait" is really just a starting point for working on the artifact trait "tree,) and completing the acquisition quest will give you the artifact power to get the first minor trait. Minor traits are either a single rank or have three ranks, and in order to progress beyond one of these three-rank traits in the tree, you'll have to unlock all three ranks.
Most minor traits do simple things, like boosting the damage of Crusader Strike by 5/10/15%. Some do somewhat more complex things, but minor traits generally give you a simple percentage boost on one of your abilities.
In addition to your inherent ability, each artifact also has three major traits, which have more profound effects, such as a Balance Druid trait that gives your Starsurge a chance to summon Goldrinn to attack your foe. These major traits will be farther out along the trait tree, and are all a single rank.
Finally, if you've filled out all of your traits, you'll be able to invest any excess artifact power you acquire into a 50-rank trait that gives you some flat bonus like increased damage (at a very gradual rate.)
My understanding is that artifact power is only acquired for the weapon you're currently using, though I'm hoping that's not the case (as I'd like to quest as Retribution but dedicate my artifact power to boosting my Protection artifact.)
While these traits are certainly going to add up to a significant increase in power, there's another very important aspect to upgrading your artifact: Relics.
Each artifact weapon starts at item level 750 (I believe the max item level for weapons out of Mythic Hellfire Citadel is 725, which is also the max upgrade for invasion weapons, so they clearly want everyone to ditch their old stuff.) But obviously, over the course of the expansion we'll want our weapons to keep up with our gear. Thus: Relics.
Each artifact weapon has three relic slots - two that are available from level 100, and one that unlocks at level 110. Relics come in several varieties - there are Life, Holy, Storm, Frost, Iron, Blood, Fel, Shadow, Fire, and Arcane. Each artifact will have a mix of relic slots - for example, Ashbringer has two Holy slots and one Fire slot. The Doomhammer has one Fire, one Iron, and one Storm relic slot.
Relics will increase the item level of your artifact, such that when you get a full set of relics from, say, the Emerald Nightmare, your weapon should be equivalent to the other pieces of gear from that raid.
Relics will also increase the ranks of one of your minor traits. For example, if you have the Sunflare Coal plugged into your Doomhammer, not only will it raise the item level of the weapon by 51 points, but it will give you a fourth rank of "Forged in Lava." Normally, at three ranks Forged in Lava makes your Lava Lash 10% stronger, but this fourth rank will increase that bonus to 13%.
My general understanding is that these trait increases are mainly to use when you need to break a tie between two equivalent relics. For example, taking Exothermic Core instead of that Sunflare Coal will still increase the Doomhammer's item level by the same amount, but instead of boosting Lava Lash you'll increase the bonus to Maelstrom generation of your Rockbiter ability.
Relics will essentially be the weapon drops of the expansion, but I also believe that once socketed, replacing them will destroy them. So if you get some nice Holy Relic that you put into your Truthguard (Protection Paladin artifact shield,) you won't be able to swap it into your Ashbringer when you get something better.
Cosmetically, artifact weapons will each have several variations. At Legion's launch there will be four models (and a fifth hidden one,) and each variant model (including the original) will have four color variations. Unlocking each model and color will require various tasks, and it is highly likely that over the course of the expansion, we're going to see new models introduced.
If you can't stand your artifact's look no matter how many appearances you unlock, you will still be able to transmog over it. However, it will still be limited by weapon type. Fire Mages, for example, get a one-handed sword and an off-hand item, so if you like some staff model on your fire mage, you're a bit out of luck.
That said, there are some appearances that seem to change the weapon type - such as a variation on the Beast Mastery gun that turns it into a bow - so perhaps Blizzard will find a way to loosen transmog restrictions.