"A particle that doesn't exist yet"
Technically a particle system that is theorized to exist, but not yet isolated:
So does that imply a new type of particle in the usual sense (electrons, positrons, neutrinos etc) or in some more abstract sense?
The anyon particle is a more abstract type called a quasi-particle. An anyon would be an isolatable effect in a *real* system, where the system is constrained in a specific way. As a way of analogy, if you had an electron travelling through a semiconductor, you can think as a really complicated system of electrons travelling thorough a sea/lattice of nuclei, or you can think of it as a quasi-electron quantum particle travelling in a more homogeneous media. An Anyon is simply a ordinary particle constrained by the system to a two-dimensional system in a specific way. A quasi-particle is mostly just a mathematical apparatus to simply solving a many-body quantum-mechanical problem using a quantum-field theory approach (rather than a quantum N-body approach).
And by "theorised to exist", again, does that mean "predicted by some postulated variant of string theory / supersymmetry / unified theory of everything (like, say, sparticles or phonons)" or "follows from standard, well tested theory, but not yet observed in the wild (eg the postulated island of stability for very isotopes with very heavy nuclei)"?
Well phonons are also quasi-particles, and they doesn't require string theory or supersymmetry or unified field theory. All phonons do is describe vibrations in crystal lattices (a real phenomena). You can think of the vibrations as quasi-particles that have "mass" and "inertia", "boundary-conditions", etc, but at the end of the day the math describes a real measurable phenomena in the crystal lattice. It's just a different way to do the math.
Anyons are quasi-particles that should mathematically occur in *real* 2-dimensional constrained systems (e.g., crystal surfaces, graphene layers, etc). Kind of like there are fermions and bosons particles which have different statistical properties which cause systems to *get-weird* in different ways when you reduce the degrees of freedom (you can google about cooling a bose-einstein condensate), these 2D anyons mathematically occur in two statistical types: Abelian and non-Abelian. Anyon quasi-particles obeying Abelian statistics have *already* been observed in nature and are key to understanding the fractional quantum hall effect.
Discovery of a system with anyon quasi-particles obeying non-Abelian statistics would be key in creating so-called "braids" which are hypothesized to be a much more stable quantum system from which to implement a quantum computer rather than using spin or polarization (which is what most people use now and suffer from very fast quantum decoherence time).
Being a quasi-particle, physicists are not so-much discovering an anyon particle, but attempting to construct *real-world* two-dimensionally constrained quantum systems that should exhibit this mathematical property, and then experimentally verifying that it has the properties theorized. This is the part that hasn't been done yet.