The gedwëy ignasia (meaning "shining palm" in the Ancient Language), also called argetlam (meaning "silver hand"), was the mark of a Dragon Rider, shining brightly whenever the Rider used his magic. This was a direct result of the Rider-Dragon Bond.
It appeared as a silver diffused spiral oval marking on a Rider's palm, and would first appear after a Rider first touched his dragon. It forms with a flash of light and, as Arya described it, a sensation as though cold water is rushing through the Rider.
Generally, the hand bearing the gedwëy ignasia was the hand the Rider used to direct his magic. The mark does not necessarily have to be on the hand; it is placed on the Rider's body in whatever place they touched their dragon first, which in Eragon's case was his right hand, but it could be anywhere, such as on a Rider's neck or leg. The mark is most commonly on a Rider's hand because when they first see their dragon, they put their hand out to touch it.
Eragon's gedwëy ignasia was on his right hand, while Murtagh's was on his left. This is probably because left-handedness is traditionally a sign of wickedness, to the extent that left-handed men in medieval times could not become knights because it was believed that they were the sons of the devil; though it should be mentioned that Arya's gedwëy ignasia is also on her left hand.
To the movie adaption, the gedwëy ignasia looks like a spiral-shaped scar resembling a stylized "e".
Elva bore a gedwëy ignasia as a star mark on her forehead. It was given to her by Saphira and she has been nicknamed "Shining Brow" because of it. The spell (or "blessing") Eragon used gave her the ability to sense when she or any one around her was in danger.
Airgetlám, often anglicized as Argetlam, is directly out of the Lebor Gabala Erenn (The Books of Invasions of Ireland). In Old Irish it means "Silver Hand". This is, again, from the Ancient Language.
Airgetlám comes from the Irish story of Nuada Airgetlám, the first king of Tuatha dé Danaan. Nuada lost his arm in a battle against his rivals, which according to his people's laws made him ineligible to remain their king. Nuada relinquished the throne to Bres, who ruled for seven years. During the seven year reign of Bres, Nuada's arm was replaced by Dian Cecht with one of silver, from which he gained the epithet Airgetlám. After receiving the arm, Nuada was restored to his throne and ruled for 20 more years.
Arget, the word for "silver" in the Ancient Language, is similar to Argentum, the Latin word for "silver."