Ice

"Its ancestral sword, Ice, was forged in ancient Valyria and has been passed down through the generations."

- Bran Stark

Ice was a Valyrian steel greatsword and a heirloom of House Stark. It was used both in war and on ceremonial occasions by the Lord Stark.

Season 1
Eddard Stark uses Ice to behead Will, a deserter from the Night's Watch. Theon Greyjoy holds the scabbard, as Eddard draws the blade. He holds the sword before him as he pronounces the sentence. Later on, Eddard cleans the sword in the godswood at Winterfell.

Ser Ilyn Payne, the King's Justice, beheads Ned Stark with Ice, on the orders of King Joffrey. He keeps the greatsword afterwards.

Season 4
Ice is melted down under Tywin Lannister's orders by a Volantene blacksmith brought to King's Landing to make Jaime Lannister a new sword. However, due to Ice's substantial size, there's enough Valyrian steel to make two regular-sized longswords. When Jaime asks about the source of the Valyrian steel Tywin replies that the previous owner had no use for it. Tywin Lannister also burns Ice's wolf-pelt scabbard.

In the books
In the A Song of Ice and Fire novels, Ice is a huge and impressive weapon forged out of Valyrian steel and has been in the family's possession for over four centuries. As a Valyrian-forged weapon, it never needs sharpening. Ned uses Ice to execute the Night's Watch deserter Gared and also to sacrifice Lady.

In the books, it is actually the Qohorik blacksmith Tobho Mott who reforges Ice into two smaller swords at Tywin's order. The Free City of Qohor is famous for its blacksmiths, who still know how to reforge Valyrian steel (but not create new Valyrian steel outright). Thus while the knowledge of reforging it is rare, it isn't limited to only three men, as Tywin says in the TV series.

For unknown reason, Mott was unable to forge the two new swords in crimson as Tywin ordered, no matter how much he tried, but in mixed colors of red and the original grey.