What are distinguishable and indistinguishable particles in statistical mechanics?
On the deepest level, particles are indistinguishable if and only if they have the same quantum numbers (mass, spin, and charges).
However, in statistical mechanics one often studies effective theories where there are additional means of distinguishing particles. Two important examples:
In modeling molecular fluids, two atoms on the same molecule are distinguishable if and only if there is no molecular symmetry interchanging the two atoms, and two atoms in different molecules are distinguishable if and only if there is no congruent matching of the two molecules such that the two atoms correspond to each other.
In modeling the solid state, one typically assumes that the atoms are confined to lattice sites, and that each site is occupied at most once.. In this case, the position in the lattice is a distinguishable label, which makes all atoms distinguishable.
The computational relevance of the distinction is that permutations of (in)distinguishable particles (don't) count towards the weighting factor.
For an expanded discussion see my article at PhysicsForums.
Assume you have two particle A and B in states 1 and 2. If the two particle are distinguishable, then by exchanging the particles A and B, you will obtain a new state that will have the same properties as the old state i.e. you have degeneracy and you have to count both states when calculating the entropy for example. On the other hand, for indistinguishable particles, exchanging A and B is a transformation that does nothing and you have the same physical state. This means that for indistinguishable particles, particle labels are unphysical and they represent a redundancy in describing the physical state and that is why you would have to divide by some symmetry factor to get the proper counting of states.