(This is a very qualitative description of the link between rare regions, Griffiths effects and Bose glasses. I am happy to add details and references where needed.).
I will focus on cold atoms implementations, as this is my field.
Rare regions
Take a potential landscape with a pure, random, uncorrelated disorder:
Because of the purely random nature of the disorder, it is quite rare to have a region in which the size of the flucuations is quite small.
So rare regione = region with small to no disorder.
Anderson localisation & Bose glass
In condensed matter, a disordered potential may induce the single-particle phenomenon of Anderson localisation. This is an insulating state, but unlike the band insulator it is in a non-crystalline solid, and unlike the Mott insulator it is not a many-body/strong correlations effect. Intuitively, it is understood as the wavefunction reflecting from all the random facets of the potential landscape. If and when these reflections are of the same mangitude of the original one, it leads to destructive interference and hence spatial localisation (image):
Nomenclature
Anderson developed this theory for non-interacting particles, but it turns out that disorder-induced insulators$^\dagger$ are resilient against moderate interactions. This is called a Many-Body Localised state, or MBL.
In the literature, however, the term "MBL" has been mostly used for fermions in highly excited states. This is because Pauli blocking would intuitively place atoms in higher states, hence if localisation has reached them as well it is "more of a big deal".
The term Bose glass is more common for bosons (duh) and in their ground state. The first experimental realisation was Fallani (arxiv paper) in 1D. A 2D version was done by Choi (arxiv paper) though I would not call it a glass since they used quenches -- i.e. not probing the ground state.
Superfluid - Bose glass - Mott insulator
The Bose glass comes up in the phase diagram of bosons in the tight-binding model (Bose-Hubbard):
where these pictures are from the Fallani paper above, $U$ is the on-site energy, $J$ the next-neighbour tunnelling ("how much atoms can move around, transport") and $\Delta$ the disorder strength:
It was known that, in the absence of disorder, bosons undergo a transition from a superfluid (off diagonal long range order, phase coherence, gapless excitations) to a Mott insulator (no phase coherence, gapped excitations). This is controlled by $J/U$, i.e. superfluid when $J\gg U$ such that particles can minimise their energy by delocolasing and being long-range ordered, and an insulator when $U\gg J$ such that repulsive interactions pins the atoms on each site individually.
Adding disorder $\Delta$, introduces another phase of matter -- the Bose glass.
This is not phase coherent (like the Mott) but has gapless excitations (like the superfliud), hence it's a mixture of both. I can expand on this bit mathematically if you wish.
Phase transition: Griffits effects in the rare regions
So where do the rare regions come into play?
At the transition between a superfluid and a Bose glass.
The disorder $\Delta$ disrupts conduction over large regionsm, causing the Anderson localistaion described above.
This about being very close to the glass-superfluid transition, but from the Bose glass side. As the disorder is reduced, there are some rare regions where the disorder strength is quite small (see first picture) and which allows the wavefunction to establish "long" range order. At the transtion, these regions grow in size (percolation) and become the superfluid.
The above picture of rare regions "mediating" the transition, only applicable near the phase transition, is what is meant by Griffiths effects. Small regions can thermalise due to the decreased disorder strength, thereby disrupting MBL and recovering transport.
$^\dagger$: while zero DC conductivity is a necessary condition for disorder-induced states, it is not their defining features. The interest in these systems is that they are non-ergodic, i.e. they do not thermalise. They retain memory fo the inital conditions at long times, which makes them ideal for quantum memory and no-decoherence applications.
