This is sadly, a pretty crappy article in some respects. In particular, this quote:
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.
Bosons, however, are particles that are their own antiparticle, and they don't annihilate when they touch each other. Majorana fermions are like photons in that respect, as they act as their own antiparticles. But unlike photons, Majoranas will still annihilate when they meet their antimatter cousins.
is 100% incorrect on the difference between fermions and bosons. A bit above this, they give a partially correct explanation in saying that fermions follow the Pauli exclusion principle and bosons don't. I believe the BBC article on this was a bit better, but I don't have time to search for it right now.
Here is the correct story:
Fermions have what we call 1/2-integer spin (like 1/2, 3/2, etc), so they are always rotating in some sense. For technical reasons, this means that two identical fermions can never have the same state --- meaning two electrons can't be in the same place, for example.
Bosons have integer spin (0,1,2, etc). Two identical bosons are allowed to have the same state --- this phenomenon is what allows us to make lasers, since photons (spin 1) are bosons.
That's the difference between bosons and fermions. Note that I said nothing about antiparticles.
Now, here's the thing about antiparticles. It really comes down to the difference between real and complex numbers. A particle and its antiparticle are the same if the particle is associated with a real number. A photon can be described by real numbers, so its antiparticle is still a photon. An electron, which is a Dirac fermion like Umbran says, must be described by complex numbers, so its antiparticle, the positron, is different. A W-boson is like a photon in lots of ways, but it is a complex number particle, so it has a different type of antiparticle (the W and its antiparticle are called W+ and W-). Majorana fermions are like electrons except they can be described by real numbers, so they are their own antiparticles.
None of this really talks about annihilation of particles. In principle, a particle and its antiparticle can
always annihilate into something else. W+ and W- bosons can annihilate, for example. Photons would be able to annihilate if there were something lighter for them to annihilate into, but they're massless, so there's nothing lighter! That's why photons can't annihilate, not particle-antiparticle business.
Umbran said:
It is important to note, however, that these people did not find a new fundamental particle, but a "quasiparticle" - an arrangement in matter that you can think of acting like a particle, but isn't a physical object. In solid state physics, for example, in a system that is nearly full of electrons, you can talk about "holes" (the place where an electron ought to be, but isn't), and treat it mathematically like a particle, even when really it is the absence of a particle.
And this is a really important point that the article just glossed over.