In hadron-hadron collisions, scattering events can be classified into diffractive and non-diffractive events. I don't clearly understand the definition of this two adjectives and I don't understand if this classification is alternative with respect to the elastic vs inelastic classification or not. I mean: can an event be at the same time, say, diffractive and elastic or the words diffractive, non diffractive, elastic, inelastic refer to 4 separate classes of events? Moreover: from an experimental point of view, how can be the various classes of events identified?
Diffractive scattering is the least understood form of hadron collisions. In general you can classify hadron interactions as follows
Elastic: In elastic scattering the hadrons just change momentum, due to it being low-energy scattering and thus unable to resolve the internal structure of the hadrons.
Non-diffractive: In very-high-energy inelastic scattering the constituents of the hadrons (partons i.e. quarks and gluons) interact, which can be computed via perturbative QCD (thanks to its asymptotic freedom property). Here the hadrons are most likely destroyed by the interaction. You can identify those interactions by finding that large parts of the collision energy are found in few jets. Since removing coloured partons from hadrons would leave coloured remnants, which are not allowed to exist, a lot of particles are created between the beam direction and the jet; effectively to dress and neutralize the colour charges.
Diffractive: And between those two regimes, you have inelastic scattering which you cannot describe simply by single parton interactions. The hadrons could stay intact, but could also be destroyed. Since the energy is too low for an interaction between coloured partons, the scattering should not result in the creation of a lot of particles between the beam direction and whatever comes out of the interaction.
It is described shortly along these lines in this paper's introduction.