Here's an answer from a non-particle physicist to complement what (former) professional particle physicist Anna V has written.
"Real particles" enter and leave Feynman diagrams. Therefore, in principle, they can be detected in an experiment - they are the "terminals" of a Feynman diagram: ports through which we can "see" the system within.
In contrast, the path of a virtual particle begins and ends within a Feynman diagram. It has no "free ends" dangling over the "boundaries" of the diagram and is therefore not directly measurable. We can't detect them in experiment.
None of this is likely new to you. You're still left wondering what reality we can ascribe to virtual particles, if we can't directly detect them. You can think of virtual particles more literally as Feynman liked to do, or you might try this approach: I personally like to think of them a little more abstractly as simply as mathematical terms in a perturbation series.
A good starting point to visualise this gist is the kinds of ideas explored in the following papers:
as well as the works of the late Hilary Booth of the Australian National University.
This is not standard QED and it is very specialised and contrived: think of it as an illustrative "Baby QED" for someone (like me) who hasn't mastered quantum field theory. We consider here the system of one electron, a proton (the latter thought of as a classical particle, simply setting up an inverse square electrostatic field in a Hydrogen atom and the "virtual photons" that are swapped between them. The electron in the classical potential is of course simply described by the first quantised Dirac equation. Now we add the electromagnetic field by adding Maxwell's equations and coupling the system as follows:
$$\gamma^\mu\left(i \partial_\mu - q A_\mu\right) \psi + V \psi - \psi = 0$$
$$\partial_\nu F^{\nu\,\mu} = q\,\bar{\psi} \gamma^\mu \psi$$
$$F_{\mu\,\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$
with the Lorenz Gauge
$$\partial_\mu A^\mu = 0$$.
The first equation is the Dirac equation, the second Maxwell's equations with a charge / current (4-current) distribution determined by the probability density of the Dirac electron. The third relates the Maxwell tensor (containing the $\vec{E}$ and $\vec{B}$ fields) to the four-potential, which couples back into the Dirac equation through the "gauge covariant derivative". So we have a rather elegant, but thorny to solve, coupled nonliear system.
In the papers, the equations lead to a fixed point problem $X=F(X)$ of a certain integro-differential operator $F$, which is contractive, so the solution is the limit of the sequence:
$$X_0,\, F(X_0),\, F^2(X_0),\,\cdots$$
and can thus be solved nonperturbatively, by the contraction mapping principle and it gives an infinite series of terms corresponding to virtual pairs too. It yields an exact solution which is an infinite series, what a mathematician would call the Peano-Baker series (see Baake and Schlaegel, "The Peano Baker Series" and it is what a theoretical particle physicist would call (I believe) the Dyson series.
Now the terms in this infinite series are $X_0$: Dirac's solution of the Hydrogen atom and the higher order terms are iterated integral operators: these iterations can be thought of as the perterbations wrought by one "virtual photon", the next term involves virtual photons and virtual pair production followed by virtual pair annihilation and so forth.
The "Virtual particles" in this viewpoint can be thought of simply as an evocative "mnemonic" to the structure of the mathematical terms in the infinite series.
