Elementary particles come in two types: fermions and bosons. Fermions are particles such as electrons, leptons and quarks (which themselves make up protons and neutrons). Fermions make up matter and obey the Pauli Exclusion Principle, which says two particles can't be in the same quatum state at the same time. (This is why two protons or neutrons, for instance, can't be in the same place at once). Bosons are things such as photons and W particles, which carry forces.
Majorana fermions are so special because they are different from other fermions, which have antiparticles — particles that have the same mass but opposite charge. An electron is negatively charged, and its antiparticle is a positron. When a particle such as an electron comes into contact with its antiparticle (in this case, a positron), the two annihilate, turning into energetic photons in this example.