You probably know them better by their initials: MGM.
MGM started in 1924 with the merger of three studios, the already well-established Goldwyn Picture Co. and two lesser known studios (Metro Pictures and Louis B. Mayer Pictures), under the aegis of New York-based theater chain Loews, Inc. Mayer was installed as vice president of studio operations, and Irving Thalberg (a young producer who had gotten his start at Universal) was hired on as director of production. From the time of the merger until Thalberg's untimely death in 1936, MGM was the top studio in Hollywood, making money even during the height of the Great Depression.
After Thalberg's death, however, things started to change. Executive producers started taking charge of their own films, eventually leading to power struggles with Mayer over money and content. Mayer, whose taste in movies was decidedly conservative, supported the Hays Code and films with more general appeal than the more literary fare Thalberg preferred, and so films that were happy-go-lucky almost to the point of being sugary were more preferred by Mayer. (Thalberg generally managed to win out, however, due to his connections with Nicholas Schenck, president of Loews, Inc. Not only were his films released more often than Mayer's preferred films, but he was at one point actually paid more than Mayer.) He especially liked musicals, and eventually that became what MGM was famous for; though they had been producing musicals for several years at that point, the run of MGM classic musicals started with The Wizard of Oz in 1939.
The success MGM had during the Depression and World War II came to a halt soon after the war ended, as audiences' tastes changed. Television was taking off, and like most studios at the time, MGM was slow to acknowledge it. On top of that, the Fall of the Studio System had started, and without Loews to help tide them over when a movie flopped, MGM's finances got rockier and rockier. Before Loews spun off MGM, their management had insisted that the studio hire a "new Thalberg" to stem the tide of cost overruns; their choice, Dore Schary, ended up going head-to-head with Mayer on just about everything, particularly choices in which films to make (Schary wanted to make Darker and Edgier movies with more serious plots, whereas Mayer was content sticking to musicals and family films). Eventually, Mayer got fed up and left in 1951.
The new management still didn't help, and Schary was released from his contract after Raintree County, a big-budget American Civil War period piece, flopped. Not long after this, Loews spun the company off and closed most of the theaters, leaving it open to takeover attempts from investors. As something of a bookend to this era, the last of MGM's classic musicals, Gigi, was released in 1958.
MGM spent most of the 1960s trying to Retool itself for the new Hollywood. It made fewer pictures per year, and the ones it did make had to be huge hits because the company's future ended up depending on a hit. Huge, lavish productions were back in vogue, with MGM betting that big movies would be better suited to draw people away form their TV sets and into theaters. When it worked, it worked well (such as with the 1959 version of Ben-Hur), but flops were bound to happen (the 1962 remake of Mutiny on the Bounty was one of the biggest), and each one just added to MGM's already huge debts. Many of their once-reliable stable of actors had been let out of their contracts due to the money problems, and the studios themselves were in declining shape. However, it did do one corporate Crowning Moment of Awesome when it produced and released the critically acclaimed, taboo smashing box office hit, Blow Up, in 1966 in direct defiance of the Hays Code and thus drove the final nail into that Censorship Bureau.
Finally, a Las Vegas investor named Kirk Kerkorian bought MGM in 1969, and began a cost-cutting drive that saw MGM lose what little prestige it had left. MGM's props department was opened up and sold at auction, and its backlots were sold to housing developers. MGM filmed fewer and fewer movies at their Culver City lot, as directors in that era started moving to location shooting and cheaper rental studios. MGM also sold its distribution division to United Artists, foreshadowing the merger that would end up happening in 1981.
By the mid-1980s, MGM/UA was producing and distributing well-known movies again (such as Fame, War Games and the James Bond films), but was still racked with debt from years of poor performance (as well as UA's losses from backing Heavens Gate). Kerkorian put the studio up for sale, and Ted Turner ended up winning the bidding in 1985. However, the massive amount of debt was off-putting, and Turner decided he wanted out only a few weeks later. Turner ended up selling UA and the MGM trademark back to Kerkorian, and the studios went to Lorimar; the studio lot eventually ended up with Sony after Warner Bros bought Lorimar (Columbia Pictures had been renting half of Warners' lot from them since the early 1970s, and the Lorimar deal meant Warners could finally kick them out).
Turner kept the MGM back catalog, however, and that catalog became the backbone of his cable-TV holdings. After a fiasco with Colorization in the late 1980s, Turner eventually created Turner Classic Movies, which runs movies from the MGM and Warner Bros catalogs (which he got from UA via Associated Artists, a TV syndicator UA bought in The Sixties) uncut, uncolorized and with no commercials. Warner Bros ended up buying Turner's company in 1996, and so now the MGM catalog resides with them.
After Turner's brief involvement, MGM/UA ended up being sold to Italian financier Giancarlo Parretti, who also held French film company Pathe at the time. It turned out, however, that he had borrowed money for the deal under false pretenses, and after not being able to make the payments on the leveraged-buyout deal he'd used to buy MGM, his bank (Credit Lyonnais) foreclosed on him and took control of MGM and Pathe. Parretti ended up being convicted of securities fraud, and MGM ended up back with Kerkorian in 1997.
In 2006, MGM was bought by a consortium of investors led by Sony, with Comcast as a minor stakeholder as well. The studio was then hit with hard times yet again, which temporarily halted a certain secret agent from going on his 23rd mission and a certain little person's chance to go on The Greatest Adventure. In 2010, MGM declared bankruptcy in a deal in which the studio was turned over to its creditors, with Spyglass Entertainment assuming a management role. As the company is now purely a production company, MGM's newest productions have gone to various entities for distribution, with Lionsgate releasing The Cabin in the Woods, Film District taking on Red Dawn, Warner Bros. acquiring most rights to The Hobbit, and Skyfall, the 23rd James Bond movie, staying with Sony.
MGM is also a partial owner of the This TV digital subchannel network with Chicago broadcaster Weigel Broadcasting, which consists mostly of films and series from the MGM library (though not the portion of the library Turner purchased).
The studio's Vanity Plate, Leo the roaring lion, has been widely spoofed and has several variations of its own. On April 16, 2009, MGM celebrated its 85th anniversary.
Among the media produced in some form by MGM:
- The Birdcage
- Goodbye, Mr. Chips
- Gone with the Wind
- Grand Hotel
- The Great Ziegfeld
- James Bond
- Josie and the Pussycats (film) (non-USA)
- Meet Me in St Louis
- "The Music Box"
- Mrs. Miniver
- Our Gang
- The Pink Panther
- The Philadelphia Story
- The Secret of NIMH
- Tex Avery MGM Cartoons
- MGM Oneshot Cartoons
- Singin' in The Rain
- The Stargate Verse 
- The Tom and Jerry cartoons
- The Wizard of Oz
- The Women
- MGM was merely the U.S. distributor of the original film, not its production company.