The idea of force-carrying particles is from the
Standard Model of quantum particle physics. Force-carrying particles belong to one of the two main classifications of particles in the Standard Model, the bosons. The other major classification of particles is the fermions. In particle physics,
everything is a particle – there are no physically real entities of any other kind.
A complete answer of this question requires a thorough overview of the Standard Model and its many theoretical underpinnings, and is beyond the scope of this short post (though something you really need to acquire to gain a satisfying understanding of this question and its answer). I’ll try, however, to give as straightforward an answer to the question as I can – though please keep in mind that I’m a mere Math undergrad turned professional computer programmer, and that my academic physics is so old (early 1980s) that it only skimmed over the then very new Standard Model - in short, I’m a self-taught enthusiast, not a trained expert, so can’t explain the formal nuts and bolts of these theories the way a real physicist could.
The proton and electron (which are both fermions) in a hydrogen atom are, in a sense, constantly emitting bosons of electromagnetic attraction. This boson is the photon, exactly the same particle as that which carries visible light and other electromagnetic radiation. However, the critical difference between photons of EM radiation and those that bind the H proton to its electron is that they are
virtual, while those of EM radiation are actual. Neither particle can ever run out of them, because none are ever radiated. The photons exchanged – and the exchange must, on average, be “even”, or atoms would spontaneously violate the law of
conservation of momentum – are emitted and received only by the particles who’s interaction they carry. None “escape the system” of the interaction between the two particles.
Virtual particles are one of many features of quantum physics that are informally described as “quantum weirdness”, a description meaning that what it describes doesn’t have sensible analogies in the large-scale world of our everyday experience on which we base our intuition of what is and isn’t normal. So the next part of the explanation will likely – and until you’ve gained a comfortable familiarity with quantum mechanics, should - sound weird.
Unlike the actual photons emitted by, say, a hot gas, we can’t say with even approximate certainty how many or what the precise energies (frequencies) of the virtual photons involved in the interaction between the proton and electron are. There are many valid possible solutions to the how-many-and-what-energies question for the interaction for any particular duration – a pair of ones with exactly equal and opposite momenta, or many with total momentum equal to the change of momentum of the proton and the electron - and, quantum weirdly, even though only one of these solutions
could have happened,
all can be assumed to have, and no experiment can possible be done to determine which one actually did. That’s the definition “virtual”, and it’s offensive to the intuitive grasp of reality we’ve formed over a lifetime of observing large scale phenomena. Nonetheless, there it is, in all its weird glory.
I will be truly grateful for a clear explanation 
The Standard Model: bosons, fermions, actual and virtual particles
