When you exchange two identical particles, you normally don’t see a change (except in phase) — eg, if I have three in a row, but swap 1 and 2, then 2 and 3, you shouldn’t be able to tell that from swapping 2 and 3 then 1 and 2. The exchanges commute.

Abelian anyons are quasiparticles that when we exchange them, we get “any” phase change rather than the normal +/- 1. Hence any-on.

In non-Abelian (ie, non-commuting) systems, exchanging particles has a deeper effect on the system - which encodes a braiding group in its topology. This happens in 2+1 dimensions because the particle world lines “tangle”; since lines tangle in 3 dimensions.

https://en.wikipedia.org/wiki/Anyon