As a Druid I have 4 artefact weapons. My experience has been this:
Artefact Weapon: 1 per spec
Artefact Power: Used to "buy" Artefact Weapon "Skill Points" AKA "Traits" (http://www.wowhead.com/guides/legion/artifact-weapons#costs).
The first break point is going from 13 to 14, where this would cost MORE (6840 Artefact Power) to go from 13 to 14 than 1 to 13 COMBINED (6500 Artefact Power).
Artefact Relic: Increases Artefact Weapon ilvl.
Gives a +1 to Artefact "Skill Point", so instead of 3/3 Frenzy Regen Increase, you can get 4/4/ or 5/5 depending on your Artefact Weapon.
Each Artefact Weapon has 3 Relic slots, of which 2 are available until you complete your Class Hall Campaign after hitting 110.
Each Relic slot has a TYPE, for Resto Druid it is Life/Frost/Life.
ARTEFACT KNOWLEDGE (http://www.wowhead.com/currency=1171/artifact-knowledge)
In your class hall, once you hit 110, you can start researching Artefact Knowledge.
Costs 500 Order Hall resources per research.
can get up to 25 level of Artefact Knowledge.
Takes 5 days per level to research.
Can have 2 research workorders active at any one time.
Artefact Knowledge EXPONENTIALLY increase the Artefact Knowledge gain from an Artefact Power item. For example, a 100 AP item, when you loot with Level 10 Artfeact Knowledge will grant you 950 Artefact Power. (100+(100x850%)
Spending Artfact Power - During Levelling
Leveled as Guardian, putting all my Artefact Power into my Guardian Artefact.
I put 12 points until I got my first Golden Dragon Trait for a Guardian Druid - Embrace of the Nightmare (http://www.wowhead.com/guides/classes/druid/guardian/gear).
I will be world questing as Guardian and wont be putting any more Artefact Power into it for now.
Spending Artfact Power - At 110
I main spec Resto, so after hitting 110, I went and grabbed all 3 other Artefact Weapons (Resto, Moonkin, Feral).
I have been putting Artefact Power into my Resto Artefact Weapon ever since and have my 2nd Golden Dragon Trait (http://www.wowhead.com/guides/classes/druid/restoration/gear).
All my Artefact Power will go here from now on.
Spending Artfact Power - Offspect Artefact Weapons
Once I can get 6500 Artefact Power from 1 (ONE) item, I will get my offspec weapons to 13 points.
This wont be until AT LEAST Artefact Knowledge Level 10 which is AT LEAST 50 days of research away.
Artefact Relics
Spend these freely on your levelling Artefact Weapon.
These scale with your iLvl when dropped, so when you hit 110, you will get better ones from various sources to put into your main spec Artefact Weapon.
Artefact Relics also come in Spec Specific flavours or Change depending on Spec flavour.
So as an example, if I had an iLvl 800 Life Relic in my Resto weapon and a iLvl 790 Life Relic drops, I would likely put this into my Guardian weapon to bump that Artefact Weapon iLvl up.
Relics also provide a bonus Trait.
So as an example, if I had an iLvl 800 Life Relic in my Resto weapon added 1 Trait to increase Swiftmend by 10% (with the other 3/3 this would be 40% incraese) and a iLvl 810 Life Relic drops which added 1 Trait to increase Barkskin by 10%, I would likely not take the iLvl upgrade and see if my Guardian weapon can use this Artefact Relic.
CLEAR AS MUD?
?? HUH? HUH?
EDIT: The artefact displayed in your class hall thingy is dependant on what spec you are in as Artefact Weapons auto equip when changing specs.