Deep inelastic scattering is the name given to a process used to probe the insides of hadrons (particularly the baryons, such as protons and neutrons), using electrons, muons and neutrinos. It provided the first convincing evidence of the reality of quarks, which up until that point had been considered by many to be a purely mathematical phenomenon. It is a relatively new process, first attempted in the 1960s and 1970s. It is an extension of Rutherford scattering to much higher energies of the scattering particle and thus to much finer resolution of the components of the nuclei.
To explain each part of the terminology, "scattering" refers to the lepton (electron, muon, etc.) being deflected. 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. The "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 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.