Matter can never be destroyed, so what happens to those particles? Do they just disappear? Where does the mass go?
[Physics] What happens to a particle and antiparticle that collide
antimatterenergy-conservationmass-energyparticle-physics
Related Solutions
The do not disappear with zero energy.
Their energy (both that originating from rest mass and any kinetic energy) appears somehow. As photons, a spray of other (lighter) particles, etc.
For instance, when an electron meets a positron (that is, a anti-electron) the most common result is a pair of gamma rays each of 511 keV (in the center of momentum (CoM) frame of the $e$--$e^+$ pair). It has to be at least two because you have to conserve both energy and momentum (and angular momentum, too, but that is a complication we will ignore for the moment), and that also tells us that there have opposite momenta in the CoM frame.
It depends on your definition of annihilation. But microscopically all processes are described by Feynman diagrams such as these
of which last one describes electron positron annihilation (if it weren't for the typo in the out-going photon). But as you can see it's all a simple matter of how you turn your head around and the very same diagram represents emission (or absorption) of photon by electron (or positron). Does electron annihilate with photon and create a brand new electron? You can certainly interpret it that way. In other words, it's just a question of terminology and interpretation. Actual physics doesn't depend on how you call the process. It is encoded in the underlying math of quantum field theory (QFT).
In any case, the punchline is that annihilation (in the strict sense of particle-antiparticle inelastic collision) doesn't have a special place in one's vocabulary once they learn their QFT and particle physics. It's just one particular kind of interaction. So you might as well ask which arbitrary interactions are allowed. And answer to that is: there's quite a lot of them and they are described by Standard Model. But the basic picture is that particles can be charged under certain charges: either the familiar electromagnetic, or less familiar weak and strong charges. Or in more modern language whether some families of particles form a multiplet under some gauge group. For weak force with group SU(2) you get lepton (e.g. electron-neutrino) and quark (e.g. up-down) doublets. For strong force with group SU(3) you need triplets and these are precisely the red green blue colors of quarks that you probably heard about.
In any case, for every multiplet there is a diagram like the ones above where you have two charged particles and one mediating particle between them (photon, weak bosons or gluons). Besides this you can also get funnier diagrams with e.g. three or four gluon lines. But that's it, these are all of allowed interactions of Standard Model.
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Best Answer
The following diagram and explanation from Cornell University's page A Brief Introduction to Particle Physics may be of help:
(Note, as correctly mentioned by @HDE in the comments, the term 'mini Big Bang' is a bit misleading, but the main point remains as @Jon Custer mentioned in the comments:
A bit more detail from the Cornell site: