What is a local operator in quantum mechanics? In quantum mechanics, what exactly is meant by "local" operator?
What about a "global" or a "non-local" operator? Are these the same?
Can you also also help me understand what exactly is a local perturbation and a local symmetry?
 A: If you are dealing with a multipartite state $\lvert\Psi\rangle$, then the distinction between local and non-local operations is an important one for example for the study of entanglement.
Consider for simplicity a bipartite state $\lvert\Psi\rangle$, that is, a state that can be written as $\lvert\Psi\rangle=\sum_{ij}c_{ij}\lvert i\rangle\otimes\lvert j\rangle$.
A local operation $A\equiv (A'\otimes I)$ is one that only acts on one part of the system. For example, an operation of the form
$$A\lvert\Psi\rangle=\sum_{ij} c_{ij}(A'\lvert i\rangle)\otimes\lvert j\rangle$$
is local in that it only affects the first part of the system.
As another example, an operation of the form $A'\otimes B'$ is also local, and its action on a product state $\lvert\psi\rangle\otimes\lvert\phi\rangle$ would read
$$\lvert\psi\rangle\otimes\lvert\phi\rangle\to (A'\lvert\psi\rangle)\otimes(B'\lvert\phi\rangle).$$
One notable property of local (unitary) operations is that they do not affect the entanglement (in a way that can be made precise).
Global operations are operations that are not local.
For a two-qubit system, an example of a global operation can be a CNOT, while a local operation is any single-qubit gate.
A: A local operator is one whose action only depends on the value of the wave function (and its derivatives) at a single point.  Almost all the ordinary operators one encounters are local in this sense, including $\hat{x}$, $\hat{p}$, $\hat{L}_{z}$, etc.  The opposite of "local" in this context is not "global," but rather "nonlocal."
A nonlocal operator would act in something like the following way:
$$\hat{W}\psi\left(\vec{x}\right)=\int d^{3}x'\, W\left(\vec{x},\vec{x}'\right)\psi\left(\vec{x}'\right).$$
The nonlocality comes from the fact that the value of $\hat{W}\psi$ at a point $\vec{x}$ depends on the value of $\psi$ at other points. The condition for $\hat{W}$ to be Hermitian is $W\left(\vec{x},\vec{x}'\right)=W\left(\vec{x}',\vec{x}\right)^{*}$.
The use of the terms "local perturbation" and "local symmetry" are less clearcut.  A local perturbation might be one of two things.  It might be a perturbation (added to the Hamiltonian) that is represented by a local operator.  Or it might mean a position-dependent perturbation that goes to zero at spatial infinity.  (Without further context, it is not possible to know which is meant.  I requested a monograph from the library once, since it was supposed to be about "nonlocal solitons," thinking that it would cover solitary waves with nonlocal interactions like $\hat{W}$ above.  In fact, it turned out to be a book about non-localized solitonic phenomena.)
Having a local symmetry means that the symmetry transformation will be allowed to depend on position, but there may be other conditions applied.  Some people restrict the term "local symmetry" to mean a gauge symmetry of the second kind; other people use it to mean something different.
