(written from a Production point of view)
Lucille "Lucy" Désirée Ball (6 August 1911 – 29 April 1989; age 77) was an actress and comedienne best known for her titular role on I Love Lucy. During the mid-1960s, she was the owner and chief executive of Desilu Studios and as such responsible for approving the initial production of Star Trek: The Original Series. Ball founded Desilu with her husband, Desi Arnaz in 1950. After the couple divorced in 1960, Lucy bought out Desi's interest and ownership of the studio. Soon after, she married comedian Gary Morton, who became her new business partner as Co-chairman of the Board in running Desilu, handling the studio's business and financial affairs. (Inside Star Trek: The Real Story, 1997, pp. 3-5)
When Desilu bought Gene Roddenberry's Star Trek is... pitch in April 1964, Lucille Ball – as owner of the company – became in effect the first identifiable individual legal and commercial owner of what was yet to become the Star Trek franchise. As was customary at the time in the motion picture industry, the moment Roddenberry signed his contract, he lost all rights and title to his creation, save for his "created by" credit (with financial compensation for work actually done to be individually determined on a per production contract basis). 
Star Trek-lore has it cited that it were her affinity for Star Trek creator Roddenberry and favor of the general goals of the series as reasons for the studio to persist with Star Trek after NBC rejected the original pilot, "The Cage".  Decades later, however, this turned out to be only exactly that – lore, a by Roddenberry trumped-up outright lie, perpetuated and grossly exaggerated by him on the 1970s-1980s Star Trek convention and lecture circuit. Desilu executive Herb Solow has pointed out that Roddenberry did not know Ball personally, and never has. Too preoccupied with her own show, Ball was shielded from the operational minutiae of running her studio by her loyal executive staff (the longer serving ones fiercely protective of their employer and known as "The Old Guard" or "Lucites") who were handling the minor day-to-day business of Desilu – which Roddenberry was, despite his assertions to the contrary afterwards (Star Trek and American Television, pp. 19-37). In effect Ball had, according to Solow, actually misunderstood the premise of the series she had bought at first; she was under the impression that she had bought a show that dealt with Hollywood stars traveling the South Seas for the USO, visiting fighting troops in the Pacific. Still, she did not revert her decision after she was set straight by Solow. "She may have initially misunderstood the Star Trek concept," author Marc Cushman wrote, "but TV's "wacky redhead", known for playing a character that had always had a harebrained scheme up her sleeve, had learned well from Desi Arnaz. He had been called crazy many times by Industry insiders, but always proved his critics wrong." (Inside Star Trek: The Real Story, 1997, p. 22; These Are the Voyages: TOS Season One, 1st ed, p. 39) Solow and Producer Robert Justman have at a later point in time added, "Lucy really did not understand the show; it was very foreign to her and she was watching this thing being done. We'd talk once or twice a week and she never looked away when we were over budget. She was there with the money. No interference whatsoever, in fact as I said in the book, when I gave her the first and second pilot scripts, I don't think she even read them."
Most crucially, however, it actually had been Ball who had saved Star Trek from cancellation for the very first time, but not directly after the initial "too cerebral" refusal in February 1965 by NBC as claimed by Roddenberry in later years and taken as gospel by 1970s-2000s "Trekdom". One year later, in February 1966, her small studio found itself unexpectedly confronted with the production of three expensive television properties, all brought in by Oscar Katz and Herb Solow (who were specifically hired to do so, in order to safeguard the future existence of the ailing studio), where there had only been one before, her own I Love Lucy show. Under advisement were Star Trek, the action series Mission: Impossible and the western series The Long Hunt of April Savage (also produced by Roddenberry between the first and second Star Trek pilots), and the conservative board of directors feared, not unjustified, that the studio would financially overstretch itself. Vigorously defended by Solow, not a voting board member but a firm believer in the show, and despite the fact that Star Trek was already picked up by NBC after the second pilot, "Where No Man Has Gone Before", had been produced, virtually the entire board (including her second husband Morton and her brother Fred, but save for executive Bernard Weitzman) unanimously voted to cancel Star Trek. Katz was also a voting board member, but was not present as he was already on his way out. Yet, as Chairwoman of the Board, Ball had the power to override her board, and this she did with a mere nod of her head towards Solow. "That was all Star Trek needed," as author Cushman had succinctly put it, "A nod of Lucille Ball." (These Are the Voyages: TOS Season One, 1st ed, pp. 32, 94) For all intents and purposes, and contrary to widely held beliefs in Star Trek-lore, this was factually the very first time that The Original Series came exceedingly close to cancellation, were it not for Lucille Ball, or as one of the other nay-sayers on the board, studio accountant Edwin "Ed" Holly, later conceded, "If it were not for Lucy, there would be no 'Star Trek' today."  Roddenberry, as a "mere" contracted studio employee, was not allowed to attend, and had Star Trek been cancelled, so would he have been, his assertions to the contrary afterwards – again – notwithstanding. Ironically, the fears of the board were somewhat allayed when ABC canceled April Savage, before production of the regular series was slated to start.
Solow's and Holly's account of Ball saving the Star Trek series for the first time was later corroborated by Ball's daughter, Lucie Arnaz (wife of Star Trek V: The Final Frontier actor Laurence Luckinbill), when she reconfirmed her mother's involvement at the 2006 induction of William Shatner into the Television Academy Hall of Fame, albeit in more flowery terms, "[At one point, her own studio chiefs said], "And the two most expensive shows are Mission: Impossible and Star Trek, [so] they have to go." She used to always listen to everything the dyed-print suits said. But she said, "No, I like 'em!" And they said, "They cost too much!" And she said, "But I like 'em!" So they left them!"
Incidentally, Post-Production Executive Bill Heath, nearly ended Star Trek shortly thereafter all by himself by exercising a far too strict reign over the visual effects budget. Heath, a "Lucite" (but not a voting board member), was of the belief that he was looking out for Ball's (financial) interests by denying the Star Trek production the budget needed for the effects production. His misguided penny-pinching caused Star Trek to nearly miss its television premier broadcast deadline, which would have inevitably led to its cancellation after all. (Inside Star Trek: The Real Story, 1997, pp. 99, 110, 259-261)
Though overruling her board, Ball was not insensitive to their concerns as she had already demonstrated earlier, when the second pilot struggled to finish shooting on 28 July 1965, in a, for her so typical, plucky way. Keeling over with fatigue, Director James Goldstone, Producer Robert Justman and Solow had the stage cleared for another shoot, when they noticed another broom on the stage. It was Ball, who declared to Goldstone while sweeping, "What do I have to do to get you to finish?", and to Justman and Solow, "What I won't do to get the wrap party started!" (Inside Star Trek: The Real Story, 1997, p. 85) Goldstone himself corroborated the episode in the documentary, Inside Star Trek - The Real Story, where he has added that champagne was served after they were done.
Despite her reluctance to become involved in the operational minutiae her studio, Lucille Ball became well known for her character trait of valuing moral propriety after her marriage with Arnaz (which had fallen apart partly due to Arnaz' philandering), and she expected this of her staff and employees as well. When she found out that the married Roddenberry had an illicit affair with Majel Barrett (ironically an actress she herself had hired for Desilu and having personally instructed her in one of own comedy seminars in 1957, ) she could not abide by this kind of behavior and wanted to fire the both of them on the spot, starkly contradicting his later "affinity" assertions. Ironically, Ball had wanted to do something similar, albeit at the opposite end of the spectrum, with Mission: Impossible co-stars Barbara Bain and Martin Landau when she found out that the two were actually a married couple, and wanted to fire them as she suspected a severe case of nepotism, which she could not abide by either. And indeed, this had been the additional reason for Ball for wanting to fire the future Roddenberry couple as well, as she had also became aware that he had surreptitiously sneaked an as a blonde-disguised Barrett back into the Star Trek production (as nurse Christine Chapel) against the express wishes of NBC. Through an intermediary, her personal publicist Howard McClay, Solow had a tough time to convince Ball otherwise for both Roddenberry / Barret and the Landau couple, as Mission was also produced under his aegis. (Inside Star Trek: The Real Story, 1997, p. 223; These Are the Voyages: TOS Season One, 1st ed, pp. 25-27)