There are two distinct notions of symmetry in quantum theory which does include QFT. The former regards the structure of the space of the states (or also the algebra of observables). The latter uses the former notion but concerns the dynamical evolution.
Wigner theorem regards the first type of symmetry.
A symmetry (of first type), is a map $S$ from the set of the states to the set of the states which is bijective (so that, in particular, it is physically reversible) and preserves some structure which properly defines which notion of symmetry one choose (always unrelated with the dynamical evolution). Two possibilities are Wigner symmetries and Kadison symmetries.
(a) Wigner symmetry: the space of the states is the space ${\cal S}_p$ of pure states and the bijective map $S : {\cal S}_p \to {\cal S}_p$ preserves the transition probability(see the final remark) of a pair of states.
As is well known, in the absence of superselection rules, pure states are unit vectors $\psi$ up to phases of the Hilbert space of the system. So $\psi$ and $e^{i\alpha} \psi$ represent the same pure state $$[\psi] := \{e^{i\alpha} \psi\:|\: \alpha \in \mathbb R\}\:.$$ The transition probability of two states respectively represented by unit vectors $\psi$ and $\phi$ is the non-negative number
$$P([\psi],[\phi]):=|\langle\psi|\phi\rangle|^2$$
These numbers can be experimentally measured.
A Wigner symmetry is a bijective map $$S : {\cal S}_p \ni [\psi] \mapsto [\psi']\in {\cal S}_p$$
such that
$$P([\psi],[\phi]) = P([\psi'],[\phi'])$$
for all choices of $[\psi]$ and $[\phi]$.
(b) Kadison symmetry the space of the states is now the whole space ${\cal S}$ of generally mixed states and the bijective map $S : {\cal S} \to {\cal S}$ preserves the convex structure of ${\cal S}$.
In the absence of superselection rules, generally mixed states are trace-class unit-trace positive operators $\rho : H \to H$ also known as statistical operators. If $\rho, \pi \in {\cal S}$ and $p,q \in [0,1]$ with $p+q=1$, the convex combination $p\rho+q\pi$ is still a state ($p$ and $q$ are the weights of the mixture).
A Kadison symmetry is a bijective map
$$S : {\cal S} \ni \rho \mapsto\rho \in {\cal S}$$
such that $$(p \rho+q\pi)' = p\rho' + q \pi'$$
for all choices of $\rho, \pi \in \cal S$ and $p,q \in [0,1]$ with $p+q=1$.
I stress that there are something like 6 or 7 notions of symmetry in quantum theory (some deals with the space of observables instead of that of states as I said above, for instance isomorphisms of the orthomodular lattice of elementary observables, or isomorphisms of the von Neumann algebra of observables). All these notions are however finally proved to be identical and related with the unitary or anti unitary operators in the Hilbert space of the system.
The Wigner/Kadison theorems characterize the corresponding symmetries as follows.
If $S$ is such a symmetry, there is a unitary or antiunitary (depending on the nature of $S$) linear operator $U : H \to H$ which implements $S$, in the sense that, respectively,
$$\psi' = U\psi\quad \mbox{or}\quad \rho' = U\rho U^{-1}\:.$$
Furthermore $U$ is determined by $S$ up to a phase: $U$ and $e^{i\alpha}U$ define the same symmetry and there are no other possibilities to represent $S$ (unless $\dim H=1$).
Notice that, in particular, Wigner symmetries and Kadison symmetries coincide though constructed referring to different aspects of the theory.
The notion of symmetry is therefore quite abstract, but it is always related with some concrete idea. For instance, if we perform quantum experiments staying in an inertial reference frame, we expect that all our experiments on isolated systems are translationally invariant. I meant that if I perform the experiment here or there on an isolated quantum system then, intrinsic properties, like transition probabilities of pairs of states (prepared here and there with the same procedure) must be identical. This idea automatically implies that
spatial translations must act in terms of Wigner symmetries on the quantum system
and, in turn, Wigner's theorem entails that
there must be a unitary operator (antiunitary is impossible here) which represent a given spatial translation.
This notion of symmetry is strictly entangled with that of dinamical evolution first and that of dynamical symmetry next.
Suppose that $$S_t : {\mathbb R} \in t \mapsto [\psi_t]$$ defines the evolution law of pure states of a given quantum system.
It is possible to prove from basic principles and fair mathematical assumptions that $S_t$ is a Wigner symmetry for every fixed $t$, so that it is represented by a class of unitary operators $U_t$ defined up to phases. In view of a theorem due to Bargmann, it is possible to prove that phases can be rearranged in order to satisfy $U_{t+t'}= U_tU_{t'}$ and $U_0=I$ so that $$\psi_t = U_t \psi_{0}$$
Furthermore, a natural continuity requirement (already used actually in applying Bargmann's theorem) imply $U_t\psi \to \psi$ for $t\to 0$. In view of Stone's theorem, we can write $$U_t = e^{-itH}$$
for some selfadjoint operator $H$ (defined up to additive constants changing the arbitrary phase of $U_t$) called the Hamiltonian of the system.
Here the notion of dynamical symmetry comes out. Suppose that we are studying a Wigner symmetry $S$ represented by the (anti)unitary operator $V$. And consider
$${\mathbb R} \ni t \mapsto V\psi_t$$
In other words we change the state at every time with that symmetry. The natural question is whether or not $ V\psi_t$ is still the evolution of the system with another initial state: $$VU_t\psi_0 = U_t \phi_0 \tag{1}$$
(There is a subtlety regarding the arbitrary phase in passing from the real state $[\psi_t]$ to a representative vector $\psi_t$, but I will not enter into the details here.) Taking $t=0$ in (1), we have that $V\psi_0=\phi_0$. Since $\psi_0$ is arbitrary, the requirement above is
$$V U_t = U_t V \quad \mbox{for all $t\in \mathbb R$}\tag{2}$$
I.e., the action of the symmetry $S$ commutes with the time evolution.
Requirement (2) is the definition of a dynamical symmetry.
If $S$ is in turn a continuous symmetry, a quantum implementation of Noether theorem arises, but it is far form the initial question so I stop here.
QFT in Hilbert space (there are other more abstract formulations) is nothing but a quantum theory in a specific Hilbert space (usually taken as a Fock space) and where the algebra of observables is of a specific kind: generated by elementary observables called field operators.
In the case of a real scalar field, it is a map associating smooth real-valued functions on the spacetime $f: M \to \mathbb R$ to selfadjoint (essentially selfadjoint in general) operators
$$f \mapsto \Phi(f) = \int_{M} \Phi(x) f(x) d^4x$$
The notion of symmetry simply specialises to this context, so that it is still given in terms of unitary or antiunitary operators, but also it is often defined by fixing its action on the said elementary observables.
For instance if $T_y$ is the action of spacetime translations on smearing functions
$$(T_y f)(x) = f(x-y)$$
we can assume that there is a corresponding symmetry (unitary operator) $U_y :H \to H$ whose action is compatible with
$$U_y \Phi(f) U_y^{-1} = \Phi(T_y f)\tag{3}$$
Usually there is a preferred unit vector $\Omega \in H$ (the vacuum) such that the linear combinations of vecotors $\Phi(f_1) \cdots \Phi(f_n)\Omega$ is dense in $H$. If (3) is accompanied by
$$U_y \Omega = \Omega \tag{4}$$
it is not difficult to prove that there exists, in fact, a unique unitary map satisfying both (3) and (4).
Many symmetries in QFT are constructed in that way. There are more complicated symmetries arising for instance from the fact that there is an internal Hilbert space for the fields, describing charges, or the field is not a simple scalar but a section $\Phi_a$ of some vector bundle over the spacetime so that geometric symmetries $T$ act into a more complicated way in the right-hand side of (3) (I pass to a pointwise representation of fields)
$$U_T \Phi_a(x) U_T^{-1} = \sum_b M^{(T)}_{ab}(x) \Phi_b(T^{-1}x)$$
Remark. The notion of probability transition is pervasive in Quantum Theory. If we start with a pure state state represented by the unit vector $\psi$ and we simultaneously measure a set of observables $A_1,\ldots, A_n$ which are maximally compatible, the post-measurment state is a unique state (uniqueness corresponds to "maximality") represented by the unit vector $\psi_{a_1,\ldots, a_n}$ (defined up to a phase), where $a_1,\ldots, a_n$ are the values of the outcomes the mesurments of the observables $A_1,\ldots, A_n$.
We have such a vector $\psi_{a_1,\ldots, a_n}$ for every choice of the numbers $(a_1,\ldots, a_n)$.
The probability to obtain a certain set of outcomes $(a_1,\ldots, a_n)$ when the state is represented by $\psi$ is therefore, according to the postulates of QT,
$$P(a_1,\ldots, a_n, \psi) = |\langle \psi_{a_1,\ldots, a_n} | \psi\rangle |^2\:.$$
Since, according to the axioms of QT, the state immediately after the measurement is $\psi_{a_1,\ldots, a_n}$, it is natural to call $P(a_1,\ldots, a_n, \psi)$ the transition probability from $\psi$ to $\psi_{a_1,\ldots, a_n}$, even if that probability concerns a set of outcomes of simultaneous measurments of observables.
It is also important to stress that, when assuming that every selfadjoint operator of the Hilbert space represents an observable, as is postulated in QT in the absence of superselection rules and gauge groups, it turns out that
every state vector $\phi$ can be written in the form $\phi = \psi_{a_1,\ldots, a_n}$ for a suitable (generally non-unique) choice of the compatible observables $A_1,\ldots, A_n$.
The transition probability $|\langle \phi| \psi\rangle|^2$ can be experimentally measured in terms of statistical frequencies (after having fixed a set of observables $A_1,\ldots, A_n$ for $\phi$ as above). This is the reason why Wigner decided to use it a a building block to formulate his notion of symmetry.
