A Thaumaturge is the class, and the Black Mage is the job that you get, when you get level 30 Thaumaturge and level 15 Archer. So basically when you level your Black Mage, you Thaumaturge level follows.
Thaumaturge is based on the elements, meaning, you have fire, ice and thunder spells. The way they work is actually a bit similar to being a Moonkin in WoW. You have the Lunar and Solar, even though you don't have a bar with numbers. If you use a fire spell, the cost of fire spells will increase and it will burn down you mana quickly if you spam them.
It will also give you a buff called Astral Fire, which have the explanation in game: "Aetherial balance of mind and spirit is leaning astrally. Fire-aspected spells require more MP, but do more damage. Ice-aspected spells require less MP, but do less damage. MP regeneration has stopped."
The Umbral Ice have the following explanation: "Aetherial balance of mind and spirit is leaning astrally. Ice-aspected spells more MP, but deals less damage. MP regeneration has quickened.".
I JUST noticed something. They have made a big mistake in their tooltip for the Umbral Ice. It is supposed to say ice-aspected spells, AND NOT fire-aspected. (see picture below)
This basically means:
Fire = High damage, high mana cost, no MP regeneration
Ice = Low damage, low mana cost, quick MP regeneration
So whenever you pop your cooldowns, and have intention of dealing alot of damage, fire is the way to go. And when you get low on mana, you use the ability called Transpose, which makes the current "buff" you have, switch to it's opposition.
As is says: "Swaps Astral Fire with a single Umbral Ice, or Umbral Ice with a single Astral Fire." That way you can more quickly begin burning down your MP, or quickly regenerate, so you don't have to waste time on casting two spells, in order to achieve the other spell. And also, the cast time on Transpose is instant, but has a 12 second cooldown.
If you have used a fire spell, and obtain the fire "buff", if you then use an ice/frost spell, you will remove the fire buff, and you would have to use yet another frost spell, in order to obtain the ice/frost buff.
I hope this makes sense.
Just a couple of pictures so you can see what I mean:
Astral Fire
Umbral Ice
Transpose
(You can click the pictures to get a bigger version. Just click it again, when you get to the imgur website)
Archer
Archer, is mainly focusing on being the ranged DPS, without the casting times, which gives you alot of mobility and the ability to kite easily. Though, you have only one crowd control ability as of now, which can be used in combat, but only lasts 10 seconds, and if the mob takes damage while being CC'd, it will break the effect.
Archers gets the job Bard, which gives you alot more support throughout fights. They get 4 songs, Mage's Ballad, Army's Paeon, Foe Requiem and Swiftsong (Swiftsong is the only one, that is not learned from actually becoming a bard, but it is still a song). Besides this, you get an ability called Battle Voice, which doubles the effectiveness of all shouts, besides Swiftsong.
Quick descriptions of the song:
- Swiftsong: Increases the movement speed of all nearby party members as long as they remain within hearing distance. Effect ends upon reuse or when enmity generated. Cannot be used in battle or with other songs.
- Mage's Ballad: Refreshes MP of all nearby party members while lowering own damage dealt by 20%. MP is drained while singing. Refresh effect is lost if party members move out of hearing distance, and ends upon reuse. Cannot be used with other songs.
- Foe's Requiem: Reduces all enemy elemental resistance by 10%. MP is drained while singing. Effect is lost if targets move out of hearing distance, and ends upon reuse. Cannot be used with other songs.
- Army's Paeon: Refreshes TP of all nearby party members while lowering own damage dealt by 20%. MP is drained while singing. Refresh effect is lost if party embers move out of hearing distance, and ends upon reuse. Cannot be used with other songs.
SM? What is that an abbreviation of? Summoner? If so, I have yet myself to try out Summoner, but yes, summoner is supposed to be a pet handling class, that is a combination of Arcanist (level 30) and Thaumaturge (level 15). I cannot really tell you anything about this class, as I have not tried it, but if you want some info on it, I would get in touch with Crisis from the Stream. His name is Crisis Flamegore ingame.¨
I hope this helped a little.
I just had a good moment, where I wanted to explain things with more depth. 
