What a mess! Okay, I am going to settle this (local thingy) once and for all. I will start by explaining the term in the relativistic classical field theory and then in QFT.
Classical Field Theory: Here the field variables \phi_{a}(x) are continuous functions on \mathbb{R}^{(1,3)}. The field equations and their invariants are obtained from the Lagrangian function \mathcal{L}(x) which is (quasi) invariant under the action of the Poincare group. Usually \mathcal{L}(x) is taken to be a real function of the fields \phi_{a}(x) and of their first derivatives \partial_{\mu}\phi_{a} leading (in this case) to field equations being differential equations of the second order. If \mathcal{L}(x) depends on the state of the fields in an infinitesimally small neighbourhood of the point x, i.e., on the values of \phi_{a} and of a finite number of their partial derivatives evaluated at the point x, then \mathcal{L}(x) = \mathcal{L}\left(\phi_{a}(x) , \partial_{\mu}\phi_{a}(x) \right) , is called local Lagrangian, and the corresponding theory is said to be a local field theory. In here, the term “local fields” are also used simply because the changes in the fields at a point x are determined by the properties of the fields infinitesimally close to the point x, i.e., the dynamical evolution is completely determined by the initial data (\phi (0, \vec{x}) , \partial_{t}\phi (0 , \vec{x})) and the field equation.
Exercise (1): You are given the Cauchy surface at t_{0}, \phi (t_{0}, \vec{x}) = F (\vec{x}), \ \ \ \partial_{t}\phi (t_{0} , \vec{x}) = G (\vec{x}), and the field equation \frac{\partial^{2}\phi}{\partial t^{2}} = \nabla^{2} \phi - m^{2} \phi . You are asked to reconstruct the dynamical evolution of the Klein-Gordon field.
In the opposite case when, for example, the Lagrangian has the form \mathcal{L}(x) = \int d^{4}y \ \Lambda (\phi (x) , \phi (y) , \partial \phi (x) , \partial \phi (y)) , one obtains so-called nonlocal theories which are plagued with incurable diseases.
The invariance of the local theory under continuous (symmetry) group allows us to construct an object (for example 4-vector current j^{\mu}(x)), which is a local function of \phi_{a}(x) and \partial_{\mu}\phi_{a}(x), satisfying local conservation law \partial_{\mu}j^{\mu}(x) = 0 which leads to a time-independent (constant) quantity Q = \int d^{3}x \ j^{0}(t, \vec{x}) and is called a global charge. In here, there are two reasons for adjective “global” : (1) because \frac{dQ}{dt} = 0 and the fact that it is defined by integrating out the \vec{x}-dependence of j^{0}(t , \vec{x}), but most importantly (2) because one can show that the time-independent Q generates the global symmetry group of the theory.
Quantum Field Theory: In QFT the field \varphi (x) becomes an operator-valued distribution on \mathbb{R}^{(1,3)}. This quantum field is related to an operator on an abstract Hilbert space \mathcal{H} and, in contrast to the classical field, ceases to describe the state of the system; this state is now represented by an abstract vector of the Hilbert space. It is clear why we need operators in the quantum theory, but why operator-valued distributions and not operator-valued (continuous) functions on \mathbb{R}^{(1,3)}?
That is to say that we need to understand why the quantum field, in contrast to the classical one, cannot be a continuous function in all 4 variables (x^{0} , \vec{x}). The reason is that the (free) field equation alone does not give the full characteristic of the (free) field, the missing parts are the equal-time commutation relations which, indeed, play the role of that of the initial data in the classical case. To see that, consider the simple case of KG field equation (\partial^{2} + m^{2}) \varphi (x), \ \ \ \ \ (1) together with the equal-time commutation relations [ \varphi (t , \vec{x}) , \varphi (t , \vec{y})] = 0, \ \ \ \ \ \ \ \ \ \ \ (2a)[ \varphi (t , \vec{x}) , \dot{\varphi} (t , \vec{y})] = i \delta^{3} (\vec{x} - \vec{y}). \ \ \ \ (2b) Now, look at the non-equal-time commutator [\varphi (x) , \varphi (y)] = i\Delta (x,y) \ \mbox{id}. Since, \varphi (x) satisfies the KG equation (1), it follows that (\partial^{2}_{x} + m^{2}) \Delta (x,y) = 0. And from (2) we obtain the initial conditions on \Delta \Delta (x,y)|_{x^{0} = y^{0} = t} = 0,\frac{\partial \Delta (x,y)}{\partial x^{0}}|_{x^{0} = y^{0} = t} = - \delta^{3} (\vec{x} - \vec{y}) . Since these initial conditions depend only on (\vec{x} - \vec{y}), x^{0} - y^{0} = 0 and are distributions, the same must be true for the solution \Delta (x,y).
In QFT there is a well-founded procedure in which the field distribution can be smeared out with a “good” function f(x) in such a way that it become, in general unbounded, operator acting in the Hilbert space and is linear on the space of “good” functions: \varphi (f) \equiv \int d^{4}x \ \varphi (x) f(x) .The operators \varphi (f) for all f have a common dense domain of definition \Omega \subset \mathcal{H} and we want \varphi (f): \Omega \to \Omega, viz. \varphi (f) \Omega \subset \Omega . The domain \Omega must also be stable under the action of the infinite-dimensional unitary representation U(a ,\Lambda ) of the Poincare group. Further, the linear functional f \mapsto \langle \Psi |\varphi (f) |\Phi \rangle , should be continuous for any |\Psi \rangle , |\Phi \rangle \in \mathcal{H} with respect to the topology of the spaces of “good” functions. These spaces are usually taken to be \mathcal{S}(\mathbb{R}^{4}), the space of rapidly decreasing functions in \mathbb{R}^{4}, or \mathcal{D}(\mathcal{O}), the space of compactly-supported functions on the (bounded) open set \mathcal{O} \subseteq \mathbb{R}^{4}.
Now, we come to the badly discussed issue of this thread, so please pay careful attention to the following: In the case that \mathcal{O} \subset \mathbb{R}^{4} is a finite spacetime region, we call the operator \varphi (f) = \int d^{4}x \ \varphi (x) \ f(x) , \ \ \ f \in \mathcal{D}(\mathcal{O}) , a smeared local operator if it satisfies the following axioms:
i) Poincare covariance, U^{\dagger}(a, \Lambda) \varphi_{r}(f)U(a , \Lambda) = D_{r}{}^{s}(\Lambda) \varphi_{s} (f_{(a , \Lambda)}), where f_{(a , \Lambda)}(x) = f\left(\Lambda^{-1}(x - a)\right), and D(\Lambda) is a finite-dimensional representation of the proper Lorentz group SO^{\uparrow}(1,3). In the particular case of translation U(a , 1) = e^{i a^{\mu}P_{\mu}}, we have e^{-i a^{\mu}P_{\mu}} \ \varphi (f) \ e^{i a^{\mu}P_{\mu}} = \varphi (f_{(a , 1)}) , \ \ \ f_{(a,1)} (x) = f(x - a). The infinitesimal version of this reads [iP_{\mu} , \varphi (f)] = \varphi (\partial_{\mu}f).
ii) Local (anti)commutativity [\varphi_{r}(f) , \varphi_{s} (g)] = 0, whenever \mbox{supp} \ f \in \mathcal{D}(\mathcal{O}) is spacelike with respect to \mbox{supp} \ g \in \mathcal{D}(\mathcal{O}).
The existence of the unique vacuum together with axiom (ii) allow us to prove the separating property of the vacuum with respect to smeared local fields: since the causal complement for a bounded open set \mathcal{O} is not empty, any smeared local operator \varphi (f), \forall f \in \mathcal{D}(\mathcal{O}) annihilating the vacuum is vanishing in itself, i.e., \varphi (f) |0 \rangle = 0 \ \ \Rightarrow \ \varphi (f) = 0.
So, a non-local field (whatever it is) is one that does not satisfy the above two axioms and, therefore, with respect to it the separating property of the vacuum does not hold. Also, the absence of x in the arguments of a field (i.e., the smearing procedure) does not make the field non-local.
Exercise (2): Recall the local conservation law in classical field theory \partial^{\mu} j_{\mu} = 0. In QFT the vector current j_{\mu}(x) becomes an operator-valued distribution. Let f_{T} \in \mathcal{D}(\mathbb{R}), \ f_{R} \in \mathcal{D}(\mathbb{R}^{3}) be test functions such that \int dx^{0} f_{T}(x^{0}) = 1, f_{R}(\vec{x}) = 1 for |\vec{x}|\leq R and f_{R}(\vec{x}) = 0 for |x| \geq 2R. Write down the distributional conservation law \partial^{0}j_{0}(x) + \sum_{k}^{3} \partial^{k}j_{k}(x) = 0 in terms of smeared out local fields.
