Hadrons are particles comprised of quarks that are held together by the strong force. The most common examples are protons and neutrons. Other hadrons are relatively unstable and do not normally exist outside of particle physics laboratories. Nuclei are made up of several hadrons, held together by the residual strong force (i.e. the quarks in neighbouring hadrons are attracted to each other as well).
The interest in hadrons and ions (nuclei stripped of electrons) for use in radiotherapy is due to their unique properties of dose deposition with depth. Unlike photons and electrons, which often deposit their dose over a broad area, protons deliver relatively little dose as they pass through tissue, before reaching a point known as the Bragg peak. At this point, the proton rapidly transfers its energy to the adjacent tissue, giving 10-20 times the dose at that point relative to its path of transit. The potential for using this phenmonenon to treat deep seated targets adjacent to critical structures is the main reason for pursuing proton therapy.
I feel the need mention how cool it is that in my job I can talk about Ion Facilities :)