Ice

Ice is a Valyrian greatsword used in war and on ceremonial occasions by the Lord of House Stark.

Season 1
Eddard Stark uses Ice to behead Will, a deserter from the Night's Watch.

Ser Ilyn Payne, the King's executioner, uses Ice to execute Eddard himself, on the orders of King Joffrey Baratheon.

In the books
In the 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. It is one of the most powerful and potent symbols of Stark power and is probably the house's greatest heirloom.

Ned uses Ice to execute the Night's Watch deserter Gared, and to sacrifice Lady under King Robert's orders.