What exactly is a pomeron? The term 'pomeron' was apparently important in the early stages of QCD. I can't find any reference to it in modern QFT books, but older resources sometimes refer to it offhand, and I've yet to find any explanation of what it actually is.
Old theoretical sources such as this throw up a wall of math and seem to say that a pomeron is a purely mathematical object, whose meaning is not clear to me:

The formal definition of Reggeon is the pole in the partial wave in t-channel of the scattering process. [...] Pomeron is a Reggeon with the intercept close to 1. [...] “Hard” Pomeron is a substitute for the following sentence: the asymptotic for the cross section at high energy for the “hard” processes which occur at small distances of the order of $1/Q$ where $Q$ is the largest transverse momentum scale in the process.

But old experimental sources say that the pomeron is a particle, and that its exchange explains some features of hadron scattering cross sections. That mismatch confuses me, but Wikipedia goes even further and says the pomeron has been found:

By the 1990s, the existence of the pomeron as well as some of its properties were experimentally well established, notably at Fermilab and DESY. The pomeron carries no charges. The absence of electric charge implies that pomeron exchange does not lead to the usual shower of Cherenkov radiation, while the absence of color charge implies that such events do not radiate pions.

This makes me really confused. If the pomeron has been found, how come no modern sources ever talk about it? Is it some other particle, a glueball or a meson, maybe, under a different name? Or have pomerons been ruled out? Are the cross sections they were invented to explain now well-understood? If not, why does nobody talk about pomerons anymore?

Edit: after searching around some more, I'm getting the impression that the pomeron is an 'effective' particle, the result of the exchange of one of a whole infinite family of particles that lie on a particular Regge trajectory. But what really mystifies me is that every source steadfastly refuses to say what those particles are, i.e. their quark and gluon content. This is apparently part of the spirit of the bootstrap program, where such questions are just not allowed to be asked, but shouldn't we be able to understand this in conventional QCD?
 A: Before the quark model became the standard model for particle physics, the prevailing model for elementary particle scattering was using the theory of Regge poles.
At the time (1960s) electromagnetic interactions/scatterings could be described very well with Feynman diagrams, exchanging virtual photons. The study of strong interactions tried to reproduce this successful use of Feynman diagrams ; for example there was the vector meson  dominance model :

In particular, the hadronic components of the physical photon consist of the lightest vector mesons, ρ  , ω  and ϕ  . Therefore, interactions between photons and hadronic matter occur by the exchange of a hadron between the dressed photon and the hadronic target.

The Regge pole theory used the complex plane  and Regge trajectories to fit scattering  crossections, the poles corresponding to resonances with specific spins  at the mass of the resonance but arbitrary ones off. The exchange of Regge poles ( instead of single particles) was fitted to scattering crossection data. See this plot for some of the "fits" .
At the time , when it seemed that the Regge pole model would be the model for hadronic interactions, it was necessary to include elastic scattering, i.e. when nothing happened except some energy exchanges. The Regge trajectory used for  that was called the Pomeron trajectory.

the particles on this trajectory have the quantum numbers of the vacuum.

If you really want to delve into the subject here is a reference.
With the successes of the standard model the Regge theory was no longer mainstream, but considered old fashioned.
This abstract for  , The Pomeron and Gauge/String Duality is revisiting the pomeron .
The emergence of string theories though revived the interest in regge theory and particularly the veneziano model which describes the  regge poles and considers the resonances as excitations of a string.
A: OK, here is the OP's requested side-supplement to @anna 's mainstream answer. 
Even though the OP's request is really a history of science one, I am not reluctant to post it here, as it is not just about the pendulum of fashion in the strong interactions (only now used indistinguishably with "QCD", after the latter's acceptance).
The reason you don't see discussions of "soft at heart" physics in QFT books now is because only few people, or else, or else, are working on "high s, low t" physics today. 
So, diffraction is well-described by the exchange of some ripple of the strong vacuum, a collective excitation of QCD, most believe, the Pomeron; but, basically, people look away and relegate such to the outer edges of their mental map, like  "hic sunt dracones" in renaissance maps... The closest you'd get to your proverbial "modern source" might be E. Levin's course... Chapter 2, "The Great Theorems" is a "must" summary. It is alive and well, but out of focus, and nothing else in QCD, or elsewhere, can supplant its utility.
In case you never noticed, in 1964, the year Gell-Mann wrote his 2-page quark paper (item 12), the bulk of his research and publications was on vacuum trajectories and Regge theory. This was hardly a symptom of a collective community delusion or wrong-mindedness! It's just that soft physics is hard to do. The community moved away and only the old-guard creative Russian physicists stayed in.
What actually happened is that, in the 70s and early 80s, discovering new particles and confirming short distance QCD (hard scattering, the confirmation of the tri-linear gluon coupling, quarkonia,...) revolutionized the focus of the strong interactions, and people started doing "clean" parton scattering experiments instead of messy triple-pomeron-coupling determination ones.
Lattice gauge theory does handle soft (~collective multigluon) physics, but it is best suited for hadronic spectra, matrix elements, and even illustrating the roiling of topological excitations such as instantons in the vacuum. But I don't know of any contributions of it to diffractive physics. (It hasn't even delivered on Wilson's promises to derive the effective low energy σ-model of chiral symmetry breaking out of the fundamental QCD Lagrangian.)
So the answer to your questions "why?" is because its just too hard, and harder to make trenchant experimental predictions with it, to warrant great experimental effort. The QCD vacuum is the classic hic sunt dracones area,  real and important as it might be... But, hey!, isn't confinement, as well?
A: Reggeon(Pomeron)-hadron and reggeon-reggeon(Pomeron-Pomeron) scattering can be considered as a scattering of all possible real mesons lying on the Regge trajectory on hadrons (for the Pomeron possible state is so called "glueball"). Conceptually it is similar to Hydrogen-hadron or Hydrogen-Hydrogen scattering (Hydrogen is also "reggeon" in this sense), since Hydrogen has the spectrum of states, and each of them has its own probability to scatter on a hadron or another hydrogen atom. We can of course consider Pomeron as a mathematical object, but I prefer to have clear physical interpretation. When we consider hadron-hadron scattering with Pomeron exchange, we have "Pomeron trajectory" (like Hydrogen spectrum), simply it is continued to the kinematical region t<0 (t>0 is the region of resonances, real particles on the trajectory, glueball, for example). I'm trying to explain... ;) Since my basic research subject is diffraction.
