(written from a Production point of view)
Desilu Productions, also credited as Desilu Studios, was the production company that started the Star Trek-franchise with the production of the, initially, un-aired 1965 pilot television episode "The Cage", and the Star Trek: The Original Series television series which began airing in September 1966.
Desilu Productions was formed in 1950 by Lucille Ball and her then-husband, Desi Arnaz. The name, a portmanteau of the couple's first names, was originally applied to the Ball-Arnaz ranch. The success of the television show I Love Lucy enabled Desilu to grow and expand throughout the 1950s. When RKO Pictures went bankrupt in 1957, Desilu bought its studios and location facilities. They produced a number of shows, including The Andy Griffith Show, and also lent their facilities for various other projects, such as My Favorite Martian, I, Spy, My Three Sons and The Untouchables.
By 1964, The Lucy Show was Desilu's sole self-made production. Oscar Katz and his assistant Herb Solow (soon promoted to studio creative head) was hired to search for writers with new and interesting concepts and develop them into series ideas for the studio. They contracted two ambitious writer/producers: Gene Roddenberry with Star Trek and Bruce Geller with Mission: Impossible. Both series went into production for the 1966-67 television season. (Inside Star Trek: The Real Story p 5-14)
Lucile Ball left production in the 1960s.
In 1962, Desilu signed an six-year agreement with Paramount to a show based on Paramount Pictures properties.
Desilu had a first-refusal agreement with CBS, which is why Star Trek was first pitched to that network. However, CBS refused to buy it, opting for Irwin Allen's more family-oriented series, Lost in Space instead. When CBS passed on the show, NBC was then approached. (Inside Star Trek: The Real Story)
Like most television studios during the 1950s and 1960s, Desilu had a stable of annually-contracted actors and behind-the-scenes personnel. Such talent moved from production to production as needed, and were paid a salary instead of a per-appearance fee. A notable Star Trek-related example was Majel Barrett, who began such a contracted relationship with Desilu in 1958. 
Desilu mainly operated on the facilities bought from RKO, which included the main Gower Street studio in Hollywood, next door to Paramount Pictures, where most of Star Trek was filmed (on Stages 9 and 10, which became Paramount Stage 31 and Stage 32 after the merge). It also consisted of a studio in Culver City, where the two Star Trek pilots were filmed, and the 40 Acres backlot - most famous for being "Mayberry" in The Andy Griffith Show - , which served as a filming location for many episodes.
After the breakup of the Ball-Arnaz marriage, Desilu remained successful. In 1962, Ball bought out Arnaz and became the first woman ever to run a major Hollywood studio. Her solo success continued unabated until 15 February 1967, when Ball announced she would sell Desilu to Gulf+Western. This act brought Desilu under the same parent company as its next-door neighbor – Paramount Pictures. The event was commemorated by a dramatic ceremony in which Ball cut a ribbon of film stock which had replaced a long-standing wall between the two production companies. At the time of the ceremony on 28 July 1967, "Mirror, Mirror" was being filmed. 
Star Trek was a bone of contention in the transition between Desilu and Gulf+Western. Ed Holly once recalled a post-sale conversation he had with Charles G. Bluhdorn, chairman of Gulf+Western:
"Just a week or so after the merger, when Bluhdorn had started seeing the cost figures, he called me in the middle of the night. All I heard was 'What did you sell me? I'm going to the poorhouse!' I said, 'Charlie, you must be looking at Star Trek and Mission: Impossible. Those shows are costing almost to the dollar what our projections showed they would cost. You and your people made the judgment that that was all right." 
For a brief time, Desilu continued to act as its own subdivision of Gulf+Western, but by December 1967, Gulf+Western fully merged Desilu with Paramount, being transformed into Paramount Television. This gradual transition resulted in several different forms of copyright for episodes of the second season of Star Trek. Hence, the initial episodes of the season bear a Desilu logo and copyright, while episodes of the latter half of the second season bear a Desilu logo but a Paramount copyright.
Desi Arnaz Edit
Desi Arnaz (2 March 1917 – 2 December 1986; age 69) was a singer and actor best known for his starring role on I Love Lucy. He co-founded Desilu with his wife Lucille Ball. She bought his shares of the company in 1961, three years before Star Trek joined their studio. His son-in-law, Laurence Luckinbill appeared as Sybok in Star Trek V: The Final Frontier.
- note: this list is currently incomplete
- Staff associated with the production of Star Trek: The Original Series, both prior and after the acquisition of Desilu by Gulf+Western
- John D.F. Black - Associate Producer / Story Editor (1966)
- Steven W. Carabatsos - Story Editor (1966-67)
- Wah Chang - Production Designer/Prop Builder (subcontractor/uncredited) (1964-67)
- Gene L. Coon - Producer / Writer (1966-67)
- Alexander Courage - Music Composer / Conductor
- John M. Dwyer - Set Decorator
- George Duning - Music Composer
- Irving A. Feinberg - Property Master
- D.C. Fontana - Story Editor (1967-68)
- Gerald Fried - Music Composer
- Byron Haskin - Producer (1964)
- John Jefferies - Set Designer
- Matt Jefferies - Production Designer / Art Director
- Robert Justman - Assistant Director (1964-65) / Producer (1965-68)
- Sol Kaplan - Music Composer
- Oscar Katz - Executive in Charge of Production (1964-65)
- John Meredyth Lucas - Producer/Writer (1967-68)
- Edward K. Milkis - Post-production Supervisor / Associate Producer (1966-68)
- Gregg Peters - Assistant Director (1966-67) / Unit Production Manager (1967-68)
- Fred Phillips - Makeup Artist (1964, 1966-68)
- Gene Roddenberry - Executive Producer / Writer (1964-68)
- Jim Rugg - Special Effects
- George A. Rutter - Script Supervisor
- Herbert F. Solow - Executive in Charge of Production (1965-68)
- Fred Steiner - Music Composer
- William Ware Theiss - Costume Designer
- Craig Thompson - Office Manager Post-Production
- Andrea Weaver - Women's Costumer
- ↑ Majel Barrett's biography at the Earth: Final Conflict official site.
- ↑ wbm
- ↑ 3.0 3.1 Sanders, Coyne Steven and Tom Gilbert. Desilu: The Story of Lucille Ball and Desi Arnaz. HarperCollins. 1994. 297-298.