The muon is an unstable particle which exists for only a fraction of a second (about 10-6 seconds) before decaying (usually into an electron, and electron-antineutrino, and a muon neutrino).
Muons were discovered during the study of cosmic rays by Carl Anderson in 1936. They were discovered by studying how the particles in a cosmic ray bent within an electromagnetic field. Anderson noticed that some particles bent less sharply than electrons did, which meant they must have been heavier particles (and thus harder to deviate off their original course by the same magnetic field strength).
Most muons that exist in nature occur when pions (particles that are created in the collision of cosmic rays with particles in the atmosphere) decay. Pions decay into a muon and neutrinos.