The scepter is made from a piece of a Seanchan spear. About two feet in length, it sports a green and white tassel beneath the spearhead, and has dragons carved into the remaining haft by Maidens of the Spear.
Rand acquired the Seanchan spear when it was cut off not far behind its head by a closing gateway while he and Aviendha escaped from the Seanchan continent they had accidentally found themselves on. Instead of disposing of the spear, Rand kept it as a constant reminder of the Seanchan threat. The Far Dareis Mai were the ones who carved the dragons into it, and it eventually became informally known as the Dragon Scepter.
Rand carried the scepter with him nearly everywhere he went, using it as a symbol and reminder for himself of the Seanchan and all of the other problems he had to deal with, figuratively keeping his hands full.
Most recently, during a trap laid by Semirhage to pose as the Daughter of the Nine Moons to lure in Rand, Semirhage sent a burst of power-wrought fire at him, which burnt the scepter to a crisp, along with his left hand.