Gem atelier is a feature that allow you to customize some perks (called vessels) of your skill wheel, in order to do so you must have revealed gems to insert in the vessels that you want to activate (you are given 1 lesser and 1 regular gem of each axis to replace your old perks), the bonus will be activated taking into account the amount of vessels u have unlocked (by filling it completely with promotion points as if were regular perks), if the amount of vessels unlocked fits the type of gem you obtain a Resonance bonus (+1 bonus damage/healing for regular or lesser or +2 for greater)
To explain further lets focus on the following image:
As you can Notice here, Once you unlock a vessel it will activate the gem perk, in this case +2% fire resistance, as this is a lesser gem and we have 1 unlocked vessel we are granted a "resonance bonus" of +1 dmg/healing, and the perk (2% fire resis) comes from the revealed lesser gem of that respective axis (in this case, upper left axis) we can have up to 250 revealed gems at once from where we could chose the different bonuses of that perk.
Now, there's another tricky thing about the vessels, they aren't unlocked as before that we could "chose" which bonuses we want to activate, which means that you must activate the bonuses in order, as you can see in the following image:
As you can notice in the image I'm using a Greater Gem and trying to reach the third vessel (divine empowerment perk) by filling the last perk, but instead it activates the 2nd vessel, also as I'm using a Greater gem (which have 3 perks) The resonance bonus is inactive and I'm not receiving the +2 bonus and I won't unless I unlock all the corresponding vessels.
With that said mastering the Gem atelier is about obtaining gems, revealing them (which cost 250k lesser, 1kk regular or 6kk greater) and building your own setup, take into account that a greater gem replace the bonuses of a minor one, so u don't need nor can use more than 1 gem per axis, neither you can fuse them so it's only worth in long term to unlock greater gems and use the gems that the game provides you to fill the rest of bonuses, it's also advisable to activate several vessels at first instead of trying to fill 1 single axis, 3 different resonance bonuses provide +3 damage and you only need 225 points, compared to 475 for only +2 (take into account that if you want to activate 1 revelation perk + the vessels you require at least lvl 400).
PD: I just noticed that the Awakened/Dormant legend is wrong (but is a bug from the website itself).