Dirac vs Majorana neutrinos
Every other matter particle we know has a distinct antiparticle: the electron has the positron, a quark has its antiquark. But the neutrino, being electrically neutral, raises a strange possibility. Maybe the neutrino and the antineutrino are not two different things at all, but two faces of one and the same particle. Whether that is true is one of the deepest open questions about neutrinos.
Physicists frame it as a choice between two descriptions. A 'Dirac' neutrino is the ordinary case: the neutrino and antineutrino are genuinely distinct, just as for the electron. A 'Majorana' neutrino, named after Ettore Majorana, is the exotic case where the particle is its own antiparticle — flip a neutrino's properties to make its antiparticle and you get the very same object back. The two pictures predict the same everyday behaviour, because the difference only shows up through the neutrino's tiny mass, which makes it fiendishly hard to tell them apart experimentally.
The stakes are high. If neutrinos are Majorana particles, then lepton number is not truly conserved, and a rare nuclear process called neutrinoless double beta decay should occur — which is exactly what experiments hunt for. A Majorana nature would also support an elegant idea, the seesaw mechanism, that explains why neutrino masses are so absurdly small, and could even underpin theories of why the universe contains matter rather than nothing. So far no experiment has settled the question; it remains one of the great unanswered puzzles of particle physics.
The cleanest way to decide is neutrinoless double beta decay: it can only happen if a neutrino emitted inside a nucleus can be reabsorbed as if it were its own antiparticle. Seeing this decay would prove neutrinos are Majorana; never seeing it leaves the Dirac option open.
A single rare decay could reveal whether the neutrino is its own antiparticle.
Dirac versus Majorana is not about whether the neutrino has mass — it has either way — but about the algebraic character of that mass. The names refer to two mathematically different ways a neutral particle can be massive.