I know its a weapon quality in Daggerfall, but what is it exactly?

Like how is it different from other materials?

Like, did you play Tribunal?

Adamantium is a metal, both in the TES universe and in other fictional mediums. Adamantine is an adjective used to describe an item with a hardness or luster of diamond.

So, in essense, I think Adamantium - at least, in most fictional literature - is supposed to be as 'hard as diamond.' Note Wolverine's claws and skeletal structure.

As for TES, in Tribunal there are ore deposits of Adamantium, and it has the appearance of a metal and is shaped by a blacksmith. So, I would assume that Adamantium is a metal.

Some ore or metal, which is high quality.

Its name is a reference to the mysterious, silvery metal of Adamantine Tower, which is in turn a corruption of Adamantia, meaning roughly 'Spirit-Freedom.'

I never played tribunal, only bloodmoon.

In TES III Tribunal, it's apparently a very rare ore that can be crafted into weapons or medium armor between Orcish and Indoril in quality.

Good ole Adamantium. Black as night, a bit heavier than you'd want it, but enchantable as hell. It was in Arena and Daggerfall, but I forgot what color it was in Arena. Purple I think.

Not too rare in other places though, it was all over in Arena. In Daggerfall it was harder to find, but no harder than finding a BMW in the States. Though earlier it had been a step below Mithril and Orcish was a step above Ebony.

Adamantium: a pretty universal name (across not only settings, but genres) for a really, really hard metal. Never mind Tribunal; you've never been exposed to the X-Men? It's the stuff that once coated Wolverine's bones. Dwarves make armor out of it in Dungeons and Dragons. It's the +2 metal resource found in Master of Magic. Doubtless there are countless other references to Adamantium in other settings. Surely you've encountered this word elsewhere?