According to Wikipedia, a Mig-29 (the most commonly used mig
currently) is "a 4th generation jet fighter aircraft designed for
the air superiority role in the Soviet Union. Developed in the
1970s by the Mikoyan design bureau, it entered service with the
Soviet Air Force in 1983, and remains in use by the Russian Air
Force as well as in many other nations. NATO's reporting name for
the MiG-29 is "Fulcrum", which was unofficially used by Soviet
pilots in service.[1] It was developed to counter new American
fighters such as the F-16 Fighting Falcon, and the F/A-18 Hornet"
(Wikipedia - Mikoyan MiG-29).
"Mig" is a general name of the Russian fighter aircraft. It is a
shortened name for the aircraft made by the Mikoyan company.
However over the years, "Mig" would refer to the current aircraft
that was used by the Russians. So a pilot in the Viet Nam War might
say he shot down a Mig which would refer to a Mig 21 jet. Sources
Cited "Mikoyan MiG-35." 14 Jun 2008 . The name "MiG" stems from
Mikoyan Gurivich, the aircraft designers.