To explain each part of the terminology, "scattering" refers to the deflection of leptons' (electron, muon, etc.) off of hadrons. Measuring the angles of deflection gives information about the nature of the process. "Inelastic" means that the target absorbs some kinetic energy. In fact, at the very high energies of leptons used, the target is "shattered" and emits many new particles. These particles are hadrons and, to oversimplify greatly, the process is interpreted as a constituent quark of the target being "knocked out" of the target hadron, and due to quark confinement, the quarks are not actually observed but instead produce the observable particles by hadronization. "Deep" refers to the high energy of the lepton, which gives it a very short wavelength and hence the ability to probe distances that are small compared with the size of the target hadron, so it can probe "deep inside" the hadron. Also, note that in the perturbative approximation it is a high-energy virtual photon emitted from the lepton and absorbed by the target hadron which transfers energy to one of its constituent quarks, as in the adjacent diagram.
Povh and Rosina pointed out that the term “deep inelastic scattering against nucleons” was coined when the quark substructure of nucleons was unknown. They prefer the term “quasielastic lepton-quark scattering”.