Are virtual particles only a fictive tool in equations? There is no "action at a distance" in nature. Attraction of a piece of iron by a magnet, attraction between distant electric charges of opposite sign, have to be mediated by something. The virtual particles are proposed as an explanation.
If they have an observable effect, it seems like they must exist. If so, why do we call them virtual? It can't be that they have short lifetimes, as there are short-lived particles (resonances) that we don't call "virtual."
On the other hand, some people say that the so-called virtual particles only exist on paper, or that they are just a technique to help us handle our equations.
So what is the meaning of the word "virtual" here? Do these particles really exist, or not?
 A: Virtual particles are not observable by definition. They represent "internal lines" in Feynman diagrams. For example, this diagram:

Here two electrons move toward each other, interact, then move away from each other. The external lines represent "real" electrons which we can measure/observe. The internal line here is an excitation of the electromagnetic field which we call a "virtual photon." Similarly, virtual electrons/quarks/gluons/etc. all correspond to internal lines in Feynman diagrams. So virtual particles are not observable by definition. If they were observable, they wouldn't be virtual.
A: The true mediators of forces are the quantum fields. 
A thorough discussion of virtual particles and their properties (and possible way of existence) is given in the following two recent essays of mine:
The Physics of Virtual Particles 
Misconceptions about Virtual Particles 
From the introduction to the second essay:
''virtual particles are defined as (intuitive imagery for) internal lines in a Feynman diagram. Their name derives by analogy to the external lines, which may be linked to observable stable or unstable particles. The 4-momentum vector of a virtual particle has the physical meaning of an integration variable in the integral corresponding to the diagram, and takes all possible values, making it off-shell.
States involving virtual particles cannot be created since quantum field theory has creation operators only for observable particles whose 4-momentum satisfies the mass-shell constraint. For lack of a state, virtual particles have none of the usual physical characteristics of real particles: They cannot be said to exist in space and time, have no position, no meaningful probabilities to be created or destroyed anywhere, no life-time, cannot cause anything, interact with anything or affect anything. Therefore there is also no dynamics, speed of motion, or world lines. (In physics, dynamics is always tied to states and an equation of motion. Neither exists for virtual particles.) [...]
The only way the usual dynamical language for virtual particles is justified by the theory is as purely figurative analogy in ”virtual reality”, useful for informal talk about complicated formulas and for superficial summaries in lectures capturing the imagination of the audience.
This has to be kept in mind when reading in professional scientific publications statements involving virtual particles. Otherwise many statements become completely misleading, inviting a magical view of microphysics and weird speculation, without the slightest support in theory or experiment.''
This is just the tip of an iceberg....
A: Virtual particles refer to actual, nonzero features in the quantum fields of real objects, but they are features that are not particles in many ways so you should not expect anything from their being named "particle".
Basically, the idea of virtual particles was invented as a device for when you want to hold on to the particle picture while doing quantum physics. Keep in mind, we know that actually nothing is really a particle, but rather quantum fields are the fundamental objects. We can derive particle-like motions in fields but fields also show other behaviours. If you insist on everything being somehow a particle then these other behaviours need to be recognised and treated with care. Someone decided they shall be called "virtual particles".
For example take a hydrogen atom, a bound proton and electron. There is for sure a real electromagnetic field inside the atom, holding it together. This electromagnetic field is certainly not a particle in any classical sense. With quantum fields we can choose to represent the electromagnetic field in terms of photons (i.e., using a photon-like basis of states to describe the field). But in doing so we see that the photons inside of a hydrogen atom are not like familiar radiating photons in free space but rather something else, virtual somehow.
Again, the only thing that is fundamental and real is the quantum field, which does not care about any distinctions we choose to make between real and virtual particles, or between particles and waves. Nevertheless we like to come up with funny names to help ourselves come to grips with the reality. But perhaps the term "virtual particle" is more misleading than it is helpful.
A: You can look the definition of a virtual particle in any text-book or jld:s answer, since it is just a definition.
About existence of anything, it becomes immediately an ontological problem and is subject to interpretation. On the advent of quantum mechanics, its opponents demanded reality. In 2016, reality of photons or virtual particles are sought after in StackExchange Physics.
I'd say virtual particles are as real as real particles, to the extent that it makes sense to talk about reality of physical objects.
There are a few complictions however in the matter.


*

*It is very hard to define particles with interactions. Given an electronic system with some degrees of freedom and photon system with some more degrees of freedom, the quantum evolution makes all exact single particle pictures vanish/invalid. For example removing an electron from an atom takes 5eV of energy, that energy consisted of huge amount of virtual photon interactions etc. It was not the energy of the electron only.
Particles are just approximate tools to understand quantum fields anyway. That is why all the confusion in 'What is really photon?' question and all the confusion about particle-wave dualities and such. I believe Higgs is just a resonance of a field and always virtual since it is not observed directly. But nobody says that Higgs is not real (neither do I).


There is a nice quote from Pauli:

Of your demands for a future [. . . ] field theory, the demand: “A single particle should appear as a trivialvsolution of the basic equations” (of which I know that it has been your favorite demand for months) seems rather questionable, since a charged particle is nothing trivial. (Pauli to Heisenberg, 16 July 1934)

source


*Particles propagating asymptiotially free are easier to define, since they no longer interact. Poles of their propagators are point like, giving well defined energy and momentum for example. The whole scattering theory is based on these asymptotic free particle states and whatever interactions may occur while scattering are called virtual.

*Something being hard does not make it real/virtual I don't quite buy most of the arguments about virtual particles not being real. They follow the same equations of motion (they "are"/are described by the same propagator) , so what if they are not in their mass-shell. If something behaves quantum mechanically and has due to interactions lost its single particle picture, that does not make it less real than the special asymptoic solution of the particle in free space.
And some more loose discussion
There is always "quantum state relativity" (I just invented that word) in quantum mechanics. If you consider detector or you being inside of your scattering matrix, then you become a virtual collection of propagators yourself. So what is inside and outside of a quantum system has to be defined. (Usually it is trivial in a normal experiment.) Now we define that what happends inside is called virtual and what is coming/going from/to outside is called real because we as humans still keep insisting on classical reality! Just like in 20's.
