Exactly what is Aizawa-sensei's quirk?
Aizawa-sensei's quirk is Erasure:

Shota's Quirk gives him the ability to nullify anyone's Quirk by looking at them.

The cloth like material is not a quirk nor is it quirk controlled. It is called the Capturing Weapon:

The Capturing Weapon is a cloth-like equipment made from carbon fibers and a special metal alloy. Shota Aizawa wears it around his neck at all times.
He is able to manipulate the weapon to ensnare his targets and immobilize them. He uses the weapon as a powerful combo with his Erasure Quirk and as a precaution for Mutant-type Quirk users who are immune to his nullification power.