Legendary Weapons are level 51 weapons that do elemental damage rather than physical
They can be Matter ( Lithic ), Spirit ( Tempestous ), Fire ( Pyroclasmic ), Cold ( Benthic )
Each one requires a dropped component, 3 arcanium weapon tinctures, 3 perfect fiery / vapor essence gems, an exceptional arcanium/arcanite weapon, and 40 arcanium/arcanite for weapons crafter items or 92 duskwood for fletcher items.
There are 4 dropped components available : Dust of Creation ( Matter), Vial of Heavenly Breezes ( Spirit ), Torch of Innovation ( Fire ), Cauldron of Vivifying Water ( Cold )
Or to put it in a table :

Albion and Hibernia Legendary Weapons are at weapon crafter skill 1080. Migard Legendary Weapons are at weapon crafter skill 1100.
All Legendary Weapons made by Fletchers are at fletching skill 1080.
The exceptional arcanium/arcanite weapons are at weapon crafter skill 1020.
The arcanium weapon tinctures are at alchemy skills 944, 949, 954, and 959 .
The perfect fiery or vapor essence gems are at spellcrafter skill 920.
The dropped components are dropped by various epic mobs in Trials of Atlantis.

The following note on the dropped components is from Sean Flynn:

ToA = 4 Elements of Nature (Earth, Wind , Water and Fire)

Earth = Dust of Creation (Matter) = Stygia (desert = dust)

Wind = Vial of Heavenly Breezes (Spirit) = Aerus (Aerus = Air = Breezes)

Water = Cauldron of vivifying water (Cold) = Oceanus (Ocean .. lots of Water ;-) )

Fire = torch of Innovation (Fire) = Volcanus (volcanos .. lava ,, ouch hot hot ;-) )

The following list shows the exceptional weapon (the base weapon's speed and weapon type) for each legendary weapon. In general the legendary weapon's speed is around that of the exceptional weapon - buts it not always exact. These are in order that they are in by the crafting list - so same weapon types aren't together. The letters in the last collumn indicate the types that I have screen shots / stats of. b - benthic, l - lithic, t - tempetous, p - pyroclasmic