The gedwëy ignasia (meaning "shining palm" in the Ancient Language) 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 and it was a silver diffused spiral oval marking that appeared on the Rider's palm after the Rider first touched his dragon. It forms with a flash of light and as Arya described it, it feels like cold water is rushing through you. Depending on the Rider, this could be on either the left or the right palm. Generally, the hand bearing the gedwëy ignasia was the hand the Rider used to direct his magic. It is also called Argetlam, meaning "Silver Hand". The mark does not necessarily have to be on the hand. The mark is placed wherever the dragon touched you first. In Eragon's case that was on his right hand. But it could be anywhere like your neck or your leg. The mark is commonly placed on the hand, because when the rider's first see the dragons, 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 the medieval age 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 this 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 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 peoples 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 retained the name Airgetlám. After receiving the arm, Nuada was restored to his throne and ruled for 20 more years.
Argetlam and Arget, the word for silver in the Ancient Language, are similar to Argentum which is the Latin word for silver.