Star Trek
Star Trek is a science fiction series that was created by Gene Roddenberry in the mid sixties.
Its current canon consists of 6 series, totaling 28 seasons (a total of 704 one-hour live action episodes and 22 half-hour animated episodes) and 10 movies (with an 11th in the works.) It was also the basis for many hundreds of novels, technical books, licensed products and toys.
The Series
Star Trek
Produced from 1966 to 1969 (2266-2268 in universe), Star Trek -- sometimes called TOS (The Original Series) -- is often considered the best of the existing series due to its iconic characters and villains, as well as stories written by established science fiction authors of its day. It was set aboard the USS Enterprise as the characters explored the "frontiers" of known space in a sort of analogy to the United States' own days of westward exploration as well as engaged in some Cold-War-era military confrontations.
Star Trek: The Animated Series
Produced from 1973 to 1974, the animated series continued the mission of the USS Enterprise with most of the same crew as the live-action show (minus Chekov), and many of the same writers, producers, and guest stars. It was the first Star Trek show to win an Emmy award.
Star Trek: The Next Generation
Produced from 1987 to 1994 (2364-2370 in universe), The Next Generation was set aboard the USS Enterprise-D. It has featured the highest ratings of any Star Trek series thus far.
Star Trek: Deep Space Nine'
Produced from 1993 to 1999 (2369-2375 in universe), "DS9" was set aboard a Bajoran-owned, Federation-administered space station near the newly discovered Bajoran Wormhole, as opposed to the traditional starship setting. The show often meets with mixed reactions, as some prefer the initial concept of the show being focused on the Starfleet personnel and their interactions with the recently liberated and impoverished Bajoran people, and some prefer the later Dominion War focus of the show.
Star Trek: Voyager
Produced from 1995 to 2001 (2371-2377 in universe), Voyager was set aboard the USS Voyager, which was stranded in the distant Delta Quadrant and trying to make its way back to Federation space with a crew comprised of both Starfleet and Maquis personnel. It is often regarded as a terrible show because some see it as having poor story ideas and poor consistency despite a promising premise.
Star Trek: Enterprise
Produced from 2001 to 2005 (April 16, 2151-January 22, 2155 in universe) and initially called just Enterprise, this prequel series was set before the birth of the Federation. The show was not well received, and most complaints about the show seemed to mirror those leveled at Voyager. It was the first Star Trek series since The Original Series not to run for at least seven seasons.
Films
Star Trek: The Motion Picture (1979, or 2273 in universe) Now promoted to Admiral, Kirk takes the Enterprise on a mission to intercept a strange cloud that has been destroying Federation and Klingon ships, and is on a direct course for Earth. Despite good production values and being the most financially successful of the original ten Trek films, it is generally regarded as very dull and boring.
Star Trek II: The Wrath of Khan (1982, or 2285 in universe) Khan Noonien Singh, an old enemy of Kirk's from the Original Series, commandeers a Federation starship and attempts to take revenge on Kirk, while also gaining control of the powerful and dangerous Genesis Device. In a reversal of the previous film, it managed to overcome its small budget and is often thought of as the best Trek film.
Star Trek III: The Search for Spock (1984, or 2285 in universe) Picking up at the end of the previous film, Khan has been defeated, but Spock is dead, and the Enterprise seemingly damaged beyond repair. Upon learning that Spock may still be alive, Kirk and crew steal the Enterprise and race towards the newly-formed Genesis Planet, and a deadly confrontation with the Klingons. This film generally does not attract many strong opinions one way or the other, but is somewhat well-regarded.
Star Trek IV: The Voyage Home (1986, or 2286 in universe) When a probe arrives at Earth and begins destroying the planet's ecosystem, Kirk and his exiled crew are forced to travel back 300 years to find the only thing capable of stopping the probe... humpback whales. Despite the ludicrous plot, this is one of the best thought-of Trek films, and the most successful with general audiences until the 2009 reboot.
Star Trek V: The Final Frontier (1989, or 2286 in universe) Kirk is reduced back to Captain, but given command of a new ship, the Enterprise-A. Unfortunately it's undermanned and doesn't work properly, but that doesn't stop Starfleet sending it on a mission to rescue three ambassadors, who have been abducted by a Vulcan who claims to know the secret of universal knowledge, and how to (literally) find God. Underperformed at the box office, and almost universally regarded as an extremely poor effort with little of anything in the way of redeeming qualities. It remains the most universally despised film in the franchise.
Star Trek VI: The Undiscovered Country (1991, or 2293 in universe) The explosion of a Klingon moon leaves their homeplanet devastated, and their Empire in no state to continue its war with the Federation. The Enterprise-A is tasked with escorting the Klingon chancellor to Earth for peace negotiations, but the chancellor is soon assassinated, and it emerges that there are people on both sides determined to prevent a peace treaty. The final official film for the original cast, and regarded as one of the series' better efforts.
Star Trek: Generations (1994, or 2371 in universe) A madman devises a plot to destroy entire star systems in order to gain access to an idyllic alternate universe, and when even the Enterprise-D proves powerless to stop him and his Klingon associates, Picard is forced to turn to the long-thought dead Kirk for help. Despite some interesting ideas, a confusing plot along with the seemingly pointless destruction of the Enterprise-D and the anticlimactic death of Kirk seriously undermined the film's credibility.
Star Trek: First Contact (1996, or 2373 in universe) When the Borg's latest attempt to assimilate the Federation fails, they decide to go back 310 years and prevent the Federation from ever being formed. Picard and the crew of the new Enterprise-E are accidentally dragged back with them, and try and stop the Borg. Although very successful on its release and briefly reviving interest in the then-flagging Trek franchise, its reputation has declined badly in recent years, largely due to its rewriting of the Borg concept.
Star Trek: Insurrection (1998, or 2375 in universe) Picard discovers the Federation's plan to exploit a planet whose atmosphere grants eternal youth and heals injuries rapidly. He decides that he doesn't like this plan, and forms a rebellion with his crew to protect the planet's natives. Usually derided for its questionable moral message, and an inconsequential feel, this signaled the beginning of a very long drawn-out end for the Trek franchise.
Star Trek: Nemesis (2002, or 2379 in universe) A clone of Picard somehow ends up in control of the Romulan Empire and has two aims; the first is to prevent his own death by abducting Picard and stealing all his blood, and the second is to destroy Earth. In no small part due to a suicidal release date (competing against that year's Lord of the Rings, Harry Potter and James Bond films), the film was a total bomb. To make matters worse, it even got beat on its opening weekend by a forgettable Jennifer Lopez comedy called Maid in Manhattan. Although not usually regarded as the worst Trek film in terms of quality, it is without much doubt the most hated of the Trek series for being a derivative and nonsensical rip-off of The Wrath of Khan.
Star Trek (2009) (2258 in universe) A franchise reboot which shows the meeting of the TOS crew and describes how Kirk becomes captain of the Enteprise in this alternate timeline. It was very financially successful and won the Trek franchise its first Oscar, though remains quite a divisive film.
Star Trek Into Darkness (2013, or 2259 in universe) A Starfleet operative named John Harrison carries out a terrorist attack on Starfleet Command, and Kirk is assigned to hunt him down, whereupon he discovers that there may be a conspiracy within Starfleet itself. Regarded as being about on par with the previous film, though some fans really didn't like that it lifted certain notable scenes from The Wrath of Khan.
In the Versus Community
Within the culture of internet versus debates, Star Trek is one of the old standbys due to its wide recognition and plethora of available information.