The Stigma system is rather important in Aion and will open up to you with lvl 20. Due to all the different patches and also different translations from asian sources there are a lot of outdated and wrong informations circling the net. I don't claim my rundown is 100% correct, but goes as far as i experienced and read about it.
Once your char reaches lvl 20 you'll automaticly gain a quest called No Escaping Destiny (Memory Of The Past is the Elyos version). The quest is a pain in the arse since you have to run all over the lowbie area again to visit several NPCs that will just send you on to the next one. I guess it was done so some higher level chars will give the newbies a show with gliding wings and gear. IMO it's a tad too long and half the NPCs would have been suffice. Anyway, its well worth doing it as soon as you get the quest. Once you finish the quest you will have access to two Stigma slots, get your first Stigma Stone for free and get 300 Stigma Shards. The Stigma Stones are rare (green) drops from monsters. You can freely trade them or sell them at the broker.
Tip: Check the questline for your class and which Stigma Stone you'll be granted as a reward. Not necessary to buy/trade that stone since you'll get it for free.
To equip a Stigma Stone you need to visit the Stigma Master in the capitol city. For Asmodians its Aud (quest will lead you there as well). Equipping it will cost you 4x[lvl of Stigma Stone] in Stigma Shards. The quest gives you 300 which is enough to equip/change Stigmas three times. Stigma Shards are obtained by killing enemy faction guards and NPCs in the Abyss afaik. Once you "fuse" the Stigma Stone into a slot, you'll gain the attached skill for it and can use it like any other skill. You can only equip Stigma Stones that are for your class. There was the possibility for some time to use cross-class stones, but it was removed.
Important: So far (dunno if they'll fix this) if you fuse another Stigma Stone on top of an already equipped one, you'll destroy the old stone. You can however take out Stigma Stones (and still trade them afterwards?) first and keep them.
The first three available Templar Stigma Stones are:
Divine Fury I (lvl 20) -> Divine Fury I
Useful Stigma skill to have and you get it for free (quest reward) anyway. Ups your DPS for a short amount of time.
old name: Indignation
Holy Shield I (lvl 20) -> Holy Shield I
Not that useful early on, when i tried it out during a normal fight i reflected back 184dmg (triggered 4 times in the 10 secs) to the attacking mob. As you usually don't want to get hit by that many NPCs anyway, don't pay too much for this one. I hear the later version could be useful in PvP.
Inescapable Judgment I (lvl 22) -> Inescapable Judgment I
I only leveled my Templar to lvl 21 so far, but from what i hear this Stigma is very cool. Useful in PvE and PvP - must have basicly.
old name: Stunning Strike
Note: The term "Gains Stigma" is just misleading, only refers to you having the stigma skill and nothing else.
Stigma Stones at the broker really go from a few hundred Kinah up to several hundred thousand, depending on class/skill. As you can only have two in the beginning I'll probably settle with Indignation & Stunning Strike if i can obtain them early on.
More Stigma Slots will open up later on:
lvl 20: 2 Stigma Slots
lvl 30: 3 Stigma Slots
lvl 40: 4 Stigma Slots
lvl 50: 5 Stigma Slots
With patch 1.5 there will be the addition of three advanced Stigma slots, which can be unlocked by doing certain questlines. Advanced Stigma slots can hold either new advanced Stigma Stones or the regular Stigma Stones. Advanced Stigma Stones can't be equipped into the normal slots tho and can only be equipped when certain prerequisite Stigma Stones are equipped. The picture shows a "chain" for those advanced Stigmas, the prerequisite can be one or two basic Stigma Stones. Example: To equip one of the lvl 50 advanced Stigma Stones you need to have five prerequisite stones equipped. The playerbase is still devided if the new high lvl advanced stigmas are worth equipping all the prerequisite ones. From what i saw you can start that questline at lvl 40, but can obtain the first advanced Stigma slot with lvl 45. The new Shugo Pirate dungeon is involved in this questline it seems. Advanced Stigma Stones can be purchased via Abyss points (13.750pts for lvl 45 and 38.700pts for lvl 50 ones), not sure if they can also drop from NPCs.
For more concrete Information i guess we'll have to wait for the Powerwiki to be updated.
Update: More infos about the new 1.5 Advanced Templar Stigmas here.
Note: One of the last patches lowered the Stigma Shard requirements. I think it 3xlevel now you have to pay.