Muons are from the same family of particles as electrons, which is a family of particles called leptons. The difference is that muons are about 200 times heaver than electrons and they don't live very long.
Muons have a properly called a magnetic moment, which is a measure of the magnetic field it generates, and the stronger a particle's magnetic moment the faster it will spin around its own axis.
A basic calculation using the standard model of quantum physics predicts a value of 2 for this magnetic moment. The trouble is, nature abhors a vacuum and so-called virtual particles are always popping in and out of existence. Virtual particles have a lot of the properties of real particles but they're extremely short-lived. These virtual particles can cause deviations in the muon's magnetic field and induce a sort of wobble with its spin.
Physicists can usually account for these virtual particles. If you know what real particles you've got, you know you need to account for their virtual versions. You can cancel out all the maths and go home for the day and put your feet up.
The thing is, things don't add up when it comes to the muon. This suggests there are virtual particles they can't account for; particles as yet unknown to physics.
Physicists have been measuring these muon 'wobbles' to try and pin down the discrepancy. This is called the 'g - 2 experiment'. The experiment first ran between 1997 and 2001 and the results were announced in 2006. They found that the muon's wobble was slightly larger than the predictions made by the standard model. Back then it was an exciting discovery but it wasn't pinned down accurately enough. Theoretical models have uncertainties and, coupled with experimental results that were not precise enough, it remained interesting rather than revolutionary.
Since then, theoretical models have tightened up and experimental apparatus has been improved, so more accurate measurements of the muon's deviation are available.
On April 6th, 2021 they announced they had tied this value down to an accuracy of 4.1 sigma. 'Sigma' is a measure of how likely the observed discrepancies are just due to a statistical fluke, and 4.1 sigma gives it a 1 in 40,000 chance of that. In order to claim something as a discovery, it needs to be 5 sigma (1 in 3.5 million chance). The experimental team is continuing to analyse the data in the hope of improving that 4.1 sigma value.
But it already points quite strongly to the existence of a new particle.
You may have read that this might mean they've discovered a new force of nature. Particles are divided into two major families: fermions and bosons. Fermions are what make up matter, and bosons are the particles that mediate the various forces between fermion particles. There are currently four forces: electromagnetism, which is mediated by photons (light), the strong nuclear force, which is mediated by gluons, the weak nuclear force, which is mediated by the W and Z particles, and gravity, which is mediated by the (so far undiscovered) graviton. There is also another boson particle — the famous Higgs particle. It arises in a different way and is less an actual force than a particle the mediates the so-called Higgs field that permeates space.
If the muon's wobble is due to an unknown boson particle then it could indeed be a new force of nature. All very exciting if you like this sort of thing.