The Forrester greatsword is a weapon passed down through the Lords of House Forrester. It is incredibly valuable and coveted.
The greatsword was first seen being polished by Gared Tuttle, Lord Gregor Forrester's squire. Gared express his desire to own a fine blade like it someday. During the Red Wedding, Lord Forrester fights off Frey soldiers using this sword. Severly injured, Lord Gregor commands Gared to bring the sword back to Ironrath.
While returning to Ironrath, Gared encounters Britt and two Bolton soldiers, who were attacking Gared's family. Gared fights them with the greatsword, but is clearly not accustomed to the weight. Gared can choose to kill one of the Boltons with the sword. Gared returns the sword to House Forrester upon his return to Ironrath.
With Lord Gregor killed and his eldest son Rodrik presumed dead, the sword is passed down to Ethan Forrester, who is no more than a boy and has little training in arms. Therefore the sword remained untouched.
Rodrik later carries the weapon with him when he rally's House Forrester's strength to defend Ironrath from the Whitehills. If Rodrik is chosen to die at the docks, the sword is reluctantly taken up by his brother, Asher Forrester, as the new head of the house. When the Forresters are defeated at Ironrath, the blade's fate is unknown, though likely claimed by the Whitehills.