Could someone please tell me what is a quasi-local charge? For instance, why are Brown-York charges called quasi-local?
The value of a local quantity at a particular point in spacetime can only depend on the values of other quantities at that same point in spacetime. It cannot depend on the values of those quantities at other points in spacetime.
In contrast, the value of a nonlocal quantity at a particular point in spacetime can depend on the values of other quantities at other points in spacetime. There is nothing in principle preventing a nonlocal quantity from depending on the value of other quantities at every point in spacetime, out to infinity.
Quasi-local quantities forbid some types of nonlocal behavior. In particular, a quasi-local quantity is only allowed to depend on the values of other quantities in a finite region of spacetime around that point.