The Mikoyan Design Bureau is a producer of military aircraft and was the primary supplier to the former USSR throughout the Cold War.
The Design Bureau was founded in December of 1939, at the beginning of the second World War. The initial design team, headed by A.I. Mikoyan, M.I. Gurevich, and V.A. Romodin, designed the I-200 fighter aircraft, which was renamed MiG-1 (for Mkoyan i Gurevich - "i" is "and" in Russian.) The MiG-1 was not an auspicious beginning for the Design Bureau - it suffered from poor maneuverability, armor, and armament, and had a very limited range; in fact, only 100 were produced. Nevertheless, the team set about improving their design, and soon introduced the MiG-3. The MiG-3 was an excellent high-altitude interceptor, but was often forced into other roles by the dire situation faced by the Russians during WW II. There are numerous recorded examples of MiG-3 pilots ramming German bombers after running out of ammunition, having been ordered to stop bombers "at all costs." 3,233 MiG-3s were produced and remained in service throughout the war.
In 1941, the Design Bureau was moved to Kuibyshev (Samara today) to keep it from being overrun by advancing Nazi forces. While in Kuibyshev, the bureau improved upon the MiG-3 design by replacing the engine with a more powerful model, the AM-37. However, the AM-37 was promptly discontinued, dooming the MiG-7 prototype to be both the first and last of its kind. The MiG-9, which was the bureau's attempt to mate a radial engine to the MiG-3 airframe, suffered from poor handling and excessive vibration, and only 5 were ever built.
In 1942, the Design Bureau was relocated to its present location in Moscow, and began work on more advanced aircraft types. The I-300 (also called the MiG-9 in production) was powered by Soviet copies of a BMW jet engine; it first flew in 1946 and was the Soviets' first jet-powered aircraft (the second had its first test flight 3 hours after the I-300.) Several improved version of the MiG-9 (called 'Fargo' by NATO) were produced, before the MiG-15 'Falcon' - an advanced jet aircraft armed with 3 cannon - was introduced in 1948. The MiG-15, powered by a copied Rolls Royce Nene engine, fought American F-86 Sabres over Korea, and still serve in the air forces of Albania, Cuba, and Romania. Over 18,000 were produced, more than any other jet fighter in history.
Throughout the Cold War, the Design Bureau continued to advance the MiG line, introducing the MiG-17 'Fresco' (1952), the supersonic MiG-19 'Farmer' (1956) and MiG-21 'Fishbed' (1958; MiG-21s are still being built in China as the F-7), and the MiG-23 'Flogger' (1969), which boasted improved ground attack capabilities. Indeed: a MiG-23 pilot once mistakenly ejected on takeoff; the fighter continued to fly, crossing the entirety of Poland before finally running out of fuel and crashing into a house in Holland, killing one person. This type of performance was typical of the MiG line - while the aircraft lacked sophistication by Western standards, they performed extremely well, and were very sturdy and easy to maintain. The MiG-25 'Foxbat' (1964) and MiG-27 (1973 - a pure ground-attack variant of the MiG-23, also called 'Flogger' by NATO) continued in this tradition; the high-altitude performance of the MiG-25 so alarmed the west that the superlative F-15 was introduced to offset it.
The MiG-31 'Foxhound' (a high-performance redesign of the MiG-25, introduced in 1975) and the MiG-29 'Fulcrum' are the Mikoyan Design Bureau's most recent designs. The MiG-29 is an extremely agile air superiority fighter roughly comparable to the American F-16, boasting high performance engines and a "fly-by-wire" control system; new variants are still under development. I saw a pair of MiG-29s at the 1989 Kalamazoo Air Show - they did some pretty crazy things, like climbing straight up on afterburner before cutting their engines back to an idle and literally falling backwards out of the sky, then cutting their engines back in and pulling out of the stall. This is called a 'hammerhead,' and it was quite impressive. The Blue Angels pilots at the air show said they'd never be allowed to try anything similar in their own aircraft.
The future of the Mikoyan Design Bureau and its projects are somewhat in doubt due to the economic collapse of the Soviet Union and the poorly-defined threats facing Russia in the modern world. An advanced fighter with supercruise and thrust-vectoring capabilities is reportedly under development as the MiG-42, but whether or not it will ever enter service is not known.
I used information from the Mikoyan Design Bureau and Russian Aviation Museum websites to prepare this summary. Many thanks to VT_hawkeye for correcting me on the etymology of "MiG," and to toalight for information on the MiG-15's powerplant, the MiG-17's NATO designation, and the hammerhead maneuver.