Tetsusaiga is the sword used by the Hanyou, InuYasha, in the Anime show, "InuYasha". It was originally forged by Totosai, by orders of his father, out of his Father's fang. It was mean't to Protect InuYasha, but ever since it was broken, it is now up to InuYasha to protect himself, combining the two powers of him, and his father, creating the now Bigger Tetsusaiga, which proves difficult to hold, because of its emence weight.
From what They find, In episode 7, "Tetsusaiga, the Phantom sword" It can only be used by a weilder with Kindess and mercey for humans, coming from his father's love of Inuyasha's mother. Since Shesshomaru has no Mercy for Humans, the sword would no except him.

When you eat dirt for the first time and leave your family to build a shack in the woods

The sword used by Inuyasha in the anime of the same name, it is a sword made by the fang of Inuyasha's father, a full-dog-demon. Originally is a weak, rusted blade, but depending on its wielder's will to protect someone dear, the blade becomes a powerful fang-sword.