Moose already gave some good answers, so I'm just going to start by dropping a link to a post of mine giving a basic overview of the mechanics of how magic is drawn, focused, and cast in the setting. Towards the bottom it focuses a little more on black magic.

Aetherology Overview, Magic Schools, Sources and Foci

To quickly go over the questions:
Quote Originally Posted by Aerazen View Post
Black mage they use eather from both heaven earth and man correct?
The "heaven, earth, and man" is a Far Eastern concept that is seemingly equivalent, though in Eorzea, magi just call it aether - as all things are made of aether, it's more encompassing I guess. Black Mages, specifically, draw from the wellspring of aether from the planet, Hydaelyn. As I said in the link above, a Black Mage doesn't always have to draw aether from the planet since their own personal aether (mana/anima) is usually sufficient to cast most spells, but sometimes more advanced magicks require a greater aether source.

Quote Originally Posted by Oboro
We draw power from the world around us─the heaven, the earth, and the hearts of man─and channel this power into physical form. I am told that this is similar to your concept of...ay-ther, was it? I must confess that Eorzean words and ways remain unfamiliar to me.

Quote Originally Posted by Aerazen View Post
Can they cast spell without a staff or wand?
This was another question that got brought up in response to that initial link and I had to do a follow up addendum here, but my answer is focused more towards arcanum. I agree with Moose in that it is theoretically possible, but as the staff is the magick's channel and focus of these energies, attempting advanced incantations of the black without the proper tools would most likely be unwise.

Considering the massive amounts of energy Black Mages attempt to channel through their bodies like a conduit, the simplest miscalculation during casting or mistake in the incantation could have disastrous consequences, as Ququruka found out.

Quote Originally Posted by Ququruka
Blinded by my own lust for power, I altered one of the incantations. It was... arrogant. And foolish. I watched as chaotic aether washed over my three friends, ripping their bodies asunder.

Quote Originally Posted by Aerazen View Post
What type of spells other then ice fire lighting do they use? What spells can they do that you don't do ingame?
Like Moose said, it's hard to say with certainty just what remains available in the Thaumaturge's arsenal. It's one of those questions that's come up several times. Can the conjurers still control all six elements, even though we only have access to three now? Can a thaumaturge still utilize astral and umbral damage? Personally, I take the repetitive appearance of Essences & Permutations: A Treatise of the Six Elements, a tome which used to grace Stillglade Fane, popping up in 2.0, 3.0, and the lore book as a sign that conjurers still can commune with every element. Otherwise, it seems strange that the elements aren't divided by the three Conquests and three Submissions.

An interesting thing to note about thaumaturges (and by extension modern Black Mages), is that in both 1.0 and 2.0 the thaumaturge arsenal focuses heavily upon balancing Astral and Umbral polarities. The only difference being that in 1.0, thaumaturges altered the polarities in other creatures with the Banish and Scourge spells to inflict more damage, whereas in 2.0 they only manipulate their own polarities. I go into the polarities a little more here. But I don't see any reason why a thaumaturge who has been training in the guild for many years couldn't learn to alter both.


Hope this helps! (Sorry about all the links, but after realizing there's a pretty small character limit after that paladin post, I decided it was best not to squeeze in the vast amounts of magical theory that I've already written recently on tumblr.)