(written from a Production point of view)
Saturday Night Live, or SNL for short, is an American late-night television comedy show featuring various comedy sketches, involving both a regular cast and celebrity guests.
SNL has parodied Star Trek many times over the years in addition to appearances by many Trek actors. Since its beginning in October 1975, SNL is produced for, and airs on NBC, the network that had been the original broadcaster of I AM ERROR.
"The Last Voyage of the Enterprise"
SNL first featured a Star Trek parody in 1976 entitled "The Last Voyage of the Enterprise". It starred John Belushi as Captain Kirk, Chevy Chase as Spock, and Dan Aykroyd as Dr. McCoy and the voice of Scotty, with actress Doris Powell as Uhura and SNL set designer Akira Yoshimura in the first of several apperances on the show as Sulu. The premise of the skit finds the Enterprise being pursued through space by a 20th century automobile, "owned by a company that manufactured cookies" (a play on the fact that the full name of NBC--the network that airs SNL--is the National Broadcasting Company, whereas the "cookie company" in question is Nabisco, who's name is short for National BISCUIT Company). A passenger from the vehicle, NBC executive Herb Goodman (played by host Elliott Gould), boards the Enterprise and informs the actors that Star Trek has been cancelled. Goodman has his assistant Curtis (Garrett Morris) and an NBC stage crew break down the set, while Spock (now out of character as Leonard Nimoy and hysterically bemoaning his lost job, as quoted below) frantically wandering the set looking for his ear tips that Goodman has pulled off. McCoy, Uhura and Sulu (now also having broken character and speaking as DeForest Kelley, Nichelle Nichols & George Takei) leave the set despondently. Kirk, sitting in his chair on the now-disassembled bridge, makes one final log entry (quoted below).
- "Most peculiar, captain. I can only assume that they possess some sort of weapons deactivator, in which case I shall merely render him unconscious with my famous Vulcan nerve pinch." - Chevy Chase (as Spock)
- "I don't believe it! God!!! Everybody I know loves the show when I see the show, huh? I have a contract! I have a contract! I want my... Where's my ears? I want my ears back! I want my ears back!" - Chevy Chase (as a breaking-character Leonard Nimoy)
- "I'm a doctor, not a tailor, dammit." - Dan Aykroyd (as Dr. McCoy) [a parody of McCoy's famous phase "I'm a doctor, not a..."]
- "Captain’s Log, final entry. We have tried to explore strange new worlds, to seek out new civilizations, to boldly go where no man has gone before. And except for one television network, we have found intelligent life everywhere in the galaxy. Live long and prosper... Promise! Captain James T. Kirk, SC 937-0176 CEC." - John Belushi (as James T. Kirk) [NOTE: This is the above-mentioned final log entry depicted in the photo at right; it references Shatner's Promise margarine ads, which aired around the time the sketch was made.]
"Star Trek V: The Restaurant Enterprise"
Another parody was done on December 20, 1986, when William Shatner hosted. The sketch, which posited what the plot of the fifth movie might be, finds The Enterprise bought out by the Marriott corporation and turned into a seafood restaurant. The crew is threatened when Khan Noonien Singh brings a health inspector to the restaurant, though Kirk resolves the situation by slipping the inspector a bribe. Shatner, of course, played Kirk, Kevin Nealon played Spock, Phil Hartman played McCoy, Victoria Jackson played Janice Rand (now a waitress instead of a yeoman) and Dana Carvey played Khan and voiced Scotty and Chekov. SNL cast members Nora Dunn and Jan Hooks played patrons, as did comedian Kevin Meaney, who plays a choking victim who is saved by a "Vulcan Heimlich Pinch" delivered by Nealon's Spock.
Perhaps the most memorable exchange is this one, referring to Meaney's character:
- Kirk: "Dr. McCoy, this man needs medical attention."
- McCoy: "Damnit, Jim, I'm a doctor, not a— oh, sure."
Another memorable quote is as follows:
- Spock: "I find it curious, Captain, that Khan was betrayed by the very health inspector he employed."
- Kirk: "Never underestimate the power of Human greed, Mr. Spock."
- McCoy: "What he's saying is that perhaps there are limitations to your vaunted Vulcan logic."
- Spock: "Dr. McCoy, would you do me the very great honor of eating my shorts?"
- McCoy: "What?!"
- Kirk: "Why Spock, I believe you're becoming more Human every time!"
- (Spock smiles & McCoy laughs, while Kirk flips open his communicator)
- Kirk: "Mr. Scott, full revolve!"
"Get a Life!" sketch
The most infamous Star Trek sketch was also featured during Shatner's 1986 appearance on the show. Not so much a parody, it featured Shatner as the guest of honor at a Star Trek convention. After being asked a barrage of trivial questions by the audience, Shatner finally berates them to "get a life!", telling them it was only a TV show. After finishing his rant (and being reminded of his contractual obligations by the convention manager) Shatner quickly explains the rant was a reenactment of the "evil" Captain Kirk from "The Enemy Within".
The scene apparently caused some fans to believe these were his true feeling for Star Trek fans, though he has assured them it was only a sketch. The sketch did, however, inspire the title for his book Get a Life!
In the intro to this edition of SNL, Shatner quipped that he hopes the Trekkies out there have a sense of humor or "I'm in deep trouble!"
- See: Get A Life!
Star Trek Democrats
The cold open of the March 14, 1992 episode did a combined parody of Trekkies and American Presidential campaigns, as the remaining Democratic candidates, former California governor Jerry Brown (Dana Carvey), former Massachusetts Senator Paul Tsongas (Al Franken), and Bill Clinton (Phil Hartman), visit a Star Trek convention, hoping to sway the audience (led by Chris Farley and Mike Myers) to support them by telling them what great fans they are - but when told that Leonard Nimoy has endorsed Tsongas, Clinton breaks into a rage, screaming "He no better than Shatner!!!" and breaking the podium.
"Love Boat: The Next Generation"
Yet another was done in 1994, when Patrick Stewart hosted, in a skit called "Love Boat: The Next Generation" (parodying both TNG and The Love Boat). Essentially, it involved the USS Enterprise-D crew running the "Galaxy-class Cruiseship Pacific Princess". (The sketch involved a model of the Enterprise-D with a model cruise ship as part of the saucer section.) 
Stewart, of course, played Captain Picard. Chris Farley played Riker, Rob Schneider played Data, Phil Hartman played Worf, Tim Meadows played La Forge (although he acts more like the character of Isaac from The Love Boat, with his catchphrase, "outta sight!" a double entendre towards La Forge's blindness), Julia Sweeney was Deanna Troi, Ellen Cleghorne was Guinan (acting more like Whoopi Goldberg than the character), Melanie Hutsell was regular "Love Boat" guest star Charo, Al Franken was Tog the Ferengi (Charo's estranged boyfriend), Adam Sandler was David Brenner and David Spade was Joan Rivers. Instead of Dr. Crusher, however, the sketch featured a cameo by actor Bernie Kopell, reprising his role as Dr. Adam Bricker from The Love Boat.
The show had also begun with a Trek spoof: Stewart's opening monologue had been a laughably inaccurate "tribute" to TOS. 
Rescue 911 spoof
Another semi-parody, also aired in 1994 involved a spoof of Rescue 911 which was hosted by William Shatner at the time. Michael McKean played Patrick Stewart, filling in for Shatner. Host Roseanne played a 911 operator who keeps blowing off callers. The sketch also featured a brief appearance and the end of the sketch by Tim Meadows as Geordi.
In a 2005 episode with guest host Paris Hilton, SNL featured a sketch spoofing erotic chatlines with an offer of "nerd chatlines" for fans of various franchises including Star Wars, Dungeons&Dragons, Lord of the Rings, Doctor Who, and Star Trek.
The Star Trek part featured Maya Rudolph as Candy, with a "massage license from Rigel VII", wanting "to go where no man has gone before". Candy, dressed in a TOS operations division uniform, claimed that it was the time of the Vulcan mating season of Pon farr and that she wanted to do something logical to the caller. She also stated that like the eel-bird of Regulus V she needed to find a mate or die. The caller, dressed in a TOS command division uniform, responded that he was "giving himself the Vulcan nerve pinch" and displayed the Vulcan salute.
Quinto, Pine, and Nimoy on 'Weekend Update'
On the May 9, 2009 episode, Chris Pine and Zachary Quinto appeared on the Weekend Update segment of SNL. They tried to explain that the new film fit flawlessly into Star Trek canon, though two Trekkies in the audience (one wearing a red TOS-uniform shirt with commander's stripes, the other a t-shirt with McCoy and Uhura and fake Vulcan ears) are clearly skeptical. Quinto, for instance, mentioned that the movie took the time to explain the origins of the Kolinahr ceremony as it is connected to the fascinating pon farr marriage ritual.
Finally, however, after Pine attempts to explain how the transporter uses the Heisenberg compensator, just as previous Trek transporters did, but is unable to pronounce "Heisenberg", they admitted they actually had no idea what they were talking about. They also mentioned having been harassed by angry fans and having received threats in a language they could not decipher - either Vulcan or Hebrew. Quinto mentioned having found decapitated action figures in his mailbox every morning, and Chris Pine complained about having received notes tied to rocks that were thrown at his windows, but only scratching them, not breaking them, since they didn't throw hard enough - all of which has resulted in death stares from the insulted Trekkies. Finally, they simply express hope that these fans will still come to see the movie.
In the background, a rather familiar voice assures the two that "they will come", and Leonard Nimoy appears - at which point the two previously angered Trekkies are visibly overcome with joy (combining Vulcan salutes with a rather un-Vulcanlike display of emotion) - and states that in time Chris Pine will be accepted as equal to the original Kirk, while Zachary Quinto will be viewed as "slightly less" than equal to the original Spock, but "ultimately OK".
Nimoy attempts to assure Quinto that fans will like the movie because to not like it would not be "illogical" as host Seth Meyers suggests (having interrupted Nimoy, beaming at the idea of beating Spock himself to the logic-based punchline), but rather "would make them dickheads" - a sentiment that, coming from their hero, the Trekkies in the audience heartily accept. The appearance ends with Nimoy, Pine, and Meyers doing the Vulcan salute.
On the February 28, 2015 episode, there is a skit featuring cast member Kenan Thompson as a doctor attempting to get closer to his wife -- a Trekkie -- by attending Star Trek conventions. He later goes straight from the convention to go and operate on a man while still dressed up as Lt. Worf. After various other off-color gags, he ends up causing the patient to go into cardiac arrest and die due to being unable to have the other doctors look at him with a straight face or without laughing at his ridiculous costume. It ends with NBC announcing a new show called "Worf M.D."
The skit was subsequently followed by a card paying tribute to Leonard Nimoy, who had died the preceding day.
Broadcast May 6, 2016, this sketch features host Chris Pine as Kirk Prime and is presented as a documentary by Neil deGrasse Tyson (Kenan Thompson) on a missing TOS episode. It depicts the introduction of Spocko, Spock's stereotypical Italian-American half-brother who was fathered by Sarek with a Human hostess. Tyson closes out the sketch by remarking that at least it was better than Star Trek V: The Final Frontier.
On the October 12, 2013 episode, SNL spoofed both the film Gravity and the 2013 US government shutdown. In the skit, two astronauts lost in space contact NASA to find that the only remaining employees are two janitors. One of the janitors claims her son can help rescue the astronauts because he is always watching I AM ERROR.
In a November 2015 episode of SNL, the show presented fictional auditions for the upcoming Star Wars movie Episode VII: The Force Awakens, with an introduction by J.J. Abrams. One of those auditions featured cast member Leslie Jones, in Klingon makeup, reporting that the ship had entered the Neutral zone, although the shields were down to 12%. Jones was then informed her performance was Star Trek, while the audition was for Star Wars.