The replies which suggest that the answer to "What is a quantum field?" is unclear or even open are wrong.
The impression that this could be unclear is owed to the standard textbooks sticking to the heuristics that helped Tomonaga-Schwinger-Feynman-Dyson to guess the theory many decades back, but the mathematical nature of realistic quantum field theory was completely understood by the mid 70s and further developed since. A survey of the state of the art is at
First of all it is worthwhile to realize that there is a difference between a field configuration and an observable on the space of all field configurations.
A field itself, either in classical physics or in its quantization, is simply a function on spacetime, assigning to each spacetime point the "value" of that field at that point. Or rather, more generally it is a section of a bundle over spacetime, called the field bundle. For instance if the field bundle is a spin bundle then the field is a spinor, if it is the differential form bundle, then the field is a gauge potential as for electromagnetism, etc.
Now from the Lagrangian density one obtains two things: the equations of motion as well as a pre-symplectic form on the space of all those field histories which solve the equations of motion. This is called the covariant phase space of the theory.
An observable is a function on this covariant phase space. It sends any field history to a number, the "value of that observable on that field history". But since the covariant phase space is itself a space of functions (or rather sections), a function on it is a functional.
Among these are the "point evaluation functionals", i.e. the observables whose value on a field history is the value of that field at a given point. The business about distributions is simply that on these point evaluation functionals the Peierls-Poisson bracket is not defined (only its integral kernel is defined, which is what you see in the textbooks). So one restricts to those observables which are functionals on the space of field histories on which the Poisson bracket actually closes. These are smearings of the point evaluation functionals by compactly supported spacetime functions. So then a point evaluation functional becomes a map that once a smearing function has been specified yields an observable. This way already classical point evaluation field observables are distributions: "classical observable-valued distributions".
Now all that happens in quantization, is that the pointwise product algebra of functionals on the covariant phase space gets deformed to a non-commutative algebra. It is traditional to demand to represent this algebra inside an algebra of operators on a Hilbert space, but for the most part this is a red herring. What counts is the non-commutative algebra of quantum observables. For computing the predictions of the theory, its scattering amplitudes, it is not actually necessary to represent this by operator algebra.
Anyway, whether you like to represent the non-commutative algebra of quantum observables by operators or not, in any case the result now is that a point evaluation functional is something that reads in a smearing function and then produces the corresponding observable, exhibited now as an element of a non-commutative algebra. In this way quantum observables on fields are algebra-element valued (e.g. operator-algebra element valued) distributions.
And, yes, for free fields this does yield the familiar creation annihilation operators, for details on how this works see
There is detailed exposition of these questions at
Presently this is written up to the classical story. For the quantum theory check out the site again in two months from now.