This article in New Scientist says that all matter is actually virtual particles popping in and out of existence and nothing more. is this correct?
The terminology "virtual particle" comes from quantum field theory.
Note the third word in QFT, theory. Theory means that it is a mathematical model for calculations which will, if the theory is valid, describe concrete measurements and behaviors of physical reality. The basic building block of QFT is the Feynman diagram: a mathematical prescription that will give integrals which will describe the crossection and angular distributions and phases in experiments with elementary "particles". By "paticles" we refer to the standard model particles assumed to be elementary.
Some of the lines that you see in this image are called virtual particles and some are called on mass shell particles. The two electron lines are on mass shell, the gamma, the quarks and the gluon are virtual, in this diagram. It is incomplete because although it has as input on mass shell particles the right side has only virtual ones.
The following decay of a Kaon
shows input real and output real particles, i.e. particles we can measure with our instruments and get their lifetimes, their crossections etc and check against the theory as exemplified by the Feynman diagrams.
We observe that input is a pair of quarks and output are three pairs of quarks. It is the pairs that are real. The quarks, the gluon, the W exchanged are virtual, i.e. off mass shell particles: with the quantum numbers from the table of SM particles but an off shell mass.
Now one can say that the K meson is composed of virtual particles if one makes a one to one identification of the guess, a good guess, of how the Kaon is built up, but the Kaon itself is real, as are the pions which are the end products. Real because we measure them with our instruments. Virtual particles are a very useful mathematical tool that simplifies our view of particle interactions, that is all.
In some sense asking about "is everything virtual" is like entering a clock shop with its hundreds of clocks and say " are all clocks gears". Clocks are made up with gears but are much more than gears. Thus protons are made up with quarks and a sea of gluons, all virtual particles, but a proton is much more than the virtual particles.
Edit: I will incorporate the content of my reply to the comments here.
Electrons and all leptons are real particles when on the mass shell, i.e. on external Feynman lines. The same is true for photons and Z and W bosons when on mass shell. Quarks and gluons can never be free and measured individually, because of the nature of the strong force. So quarks are always virtual and have to appear in pairs(mesons) or triplets (protons) in the external legs of the diagrams, and it is the pairs and triplets that can be on the mass shell. Gluons cannot be on their mass shell either, and their only externally measured evidence is in hadronic jets.
The nucleus is a more complicated grouping, it contains off the mass shell protons neutrons gluons photons, but all together, the nucleus is on the mass shell. when isolated. When surrounded by electrons it becomes an atom, the atom then is the one that is on the mass shell and contains electrons +nucleus off the mass shell. When observing a molecule, the molecule is on the mass shell and contains off mass shell atoms. When observing a crystal, the whole crystal is on the mass shell and the molecules are off mass shell.
All microscopic quantum mechanically defined items can be sometimes virtual and some times real ( except quarks and gluons which are always virtual). Virtual is in the contained level and real is the total group that is under measurement and observation.
There is always a real level riding on virtual particles, nuclei, atoms, molecules depending on the magnification one looks at in the microcosm. Our reality rests on nested levels of virtuality and reality ascertains itself on the level we are examining.
In the classical world we live and move in the underlying levels of virtuality are immaterial, because in addition to the nesting process described above there is also something called "decoherence"
It statistically transforms all quantum mechanical substructures to the level we measure and describe with classical physics and certainly call real.