It's not so easy to describe...here are a few of my favorite descriptions...
Perturbation theory of the standard model of particle physics produces virtual particles...
see here:
http://en.wikipedia.org/wiki/Virtual_particle
" the accuracy and use of virtual particles in calculations is firmly established, but their "reality" or existence is a question of philosophy rather than science."
This covers some of the points just posted by Tiny Tim...
In 1926, in one of the very first papers on quantum mechanics, Born, Heisenberg and Jordan presented the quantum theory of the electromagnetic field. …Born et al. gave a formula for the electromagnetic field as a Fourier transform and used the canonical commutation relations to identify the coefficients in this Fourier transform as operators that destroy and create photons, so that when quantized this field theory became a theory of photons. Photons, of course, had been around (though not under that name) since Einstein’s work on the photoelectric effect two decades earlier, but this paper showed that photons are an inevitable consequence of quantum mechanics as applied to electromagnetism.
[If you know Fourier transforms, the virtual portion is the imaginary part. The real number part is what we can observe/detect.]
What is Quantum Field Theory, and What Did We Think It Is?
Steven Weinberg-Physics Department, University of Texas at Austin
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and perhaps my favorite description, Wikipedia:
There is not a definite line differentiating virtual particles from real particles — the equations of physics just describe particles (which includes both equally). The amplitude indicating that a virtual particle exists interferes with the amplitude for its non-existence; whereas for a real particle the cases of existence and non-existence cease to be coherent with each other and do not interfere any more. In the quantum field theory view, "real particles" are viewed as being detectable excitations of underlying quantum fields. As such, virtual particles are also excitations of the underlying fields, but are detectable only as forces but not particles. They are "temporary" in the sense that they appear in calculations, but are not detected as single particles. Thus, in mathematical terms, they never appear as indices to the scattering matrix, which is to say, they never appear as the observable inputs and outputs of the physical process being modeled. In this sense, virtual particles are an artifact of perturbation theory, and do not appear in a non-perturbative treatment.
If that isn't crazy enough, consider this: You want to create real photons from virtual photons?? You can!
Just accelerate! Examples are the Unruh effect, Hawking radiation and hanging around any cosmological horizon, like a black hole horizon. Most of the 'real' early particles in the universe were created during high energy 'inflationary' era when the high vacuum energy of virtual particles were 'bound' by the cosmological horizon...and poof, out popped real particles! If this hadn't happened, our universe would be almost empty and expanding without bound and in which any life such as we know it would be impossible.
An intuitive way to think about this 'particle creation' is that virtual particle wave-forms [say via the Schrodinger wave equation] are spread out all over the place, ill defined, not detectable. But when a horizon appears, it acts like a bound, a confinement. This is analogous to confining a particle, if you are familiar with that, like positing an electron in an atom when it can have only certain energies, or like clamping the ends of a vibrating violin...clamping, bounding, the ends, forces it into quantized harmonic behavior...
