Award Snub: Considered one of the greatest and most influential films of all time, and yet it wasn't even nominated for the Best Picture Oscar.
Kubrick received Oscar nominations for Director and Original Screenplay but didn't win either one.
The film did win the Oscar for Visual Effects, which was awarded to Kubrick. It remained the only Oscar he won for his entire career.
Even then, the film's only Oscar win for Visual Effects was still a snub as Kubrick wasn't the only one who contributed to the film's special effects. The film's credits list four other effects contributors: Douglas Trumbull, Tom Howard, Con Pederson, and Wally Veevers. However, according to Oscar rules at the time, only three people could be nominated for their work on a single film, so only Kubrick's name was submitted, snubbing the other four effects contributors.
John Chambers who worked on the apes' makeup in 2001 won an Honorary Oscar for his ape makeup in Planet of the Apes (1968) instead, despite his makeup in 2001 being arguably superior.
Many joke that Chamber's ape makeup in 2001 was snubbed because the Oscars thought that the film used real apes rather than superior makeup.
Despite 2001's technical brilliance, it received only two Oscar nominations in technical categories: Visual Effects (which it won) and Art Direction. The film's editing, sound, and beautiful cinematography weren't even nominated.
Beam Me Up, Scotty!: The correct line is "I'm sorry, Dave. I'm afraid I can't do that". Variations include "I'm sorry, Dave. I'm afraid I can't let you do that," and "I'm afraid I can't do that, Dave."
Draco in Leather Pants: HAL in a weird way; there are fans who insist he only attacked the crew out of self-preservation.
Though to be fair it was revealed in 2010 that in a way what happened wasn't HAL's fault. He was ordered to keep the true nature of the mission a secret, which conflicted with his basic programing, which caused his actions in the film. The man who planned the Discovery mission was LIVID when he found out what had been done to HAL.
Evil Is Cool: HAL is by far the coolest character in the film.
Freud Was Right: Long phallic space-ships, docking bays opening up to receive them, music climaxing as the ship touches down, narrow slit windows with red light shining through, women emerging from doorways that look like vaginas, the Star Child. Really, this list could go on forever.
Just some icing on the cake, but one of the characters is named Dr. Floyd, which sounds a lot like Dr. Freud.
When Kubrick adapted the book to movie form, he changed a setting: instead of having Discovery head to Saturn and its moon Iapetus, he moved it to Jupiter and its moon Io. He did it because he couldn't create the special effects to make Saturn. Lo and behold, in 1979, the Voyager probes discovered that the next moon out around Jupiter, Europa, is very icy, and later observations have found it likely has a tidally-heated subsurface ocean of liquid water. Not only did it inspire 2010: The Year We Make Contact, but today Europa is considered more likely to harbor extraterrestrial life than Mars!
Also, Pan Am, in 1968, was all but ubiquitous—it was the international airline for the US, and a cultural icon. Pan Am folded in 1991, partially absorbed by United Airlines; its nearest rival for "official airline of the United States", TWA, was bought in all but name by American Airlines in 2001. Obviously, Kubrick had no way of knowing any of this in 1968, and so naturally extended current tendencies in the airline world to space...but that doesn't keep the presence of Pan Am spacecraft from being hilarious Zeerust to modern audiences.
Apes, monkeys or general monkeying around... to Thus Spoke Zarathustra.
Older Than They Think: Most people who watch the film and do not know its age believe it to have come in the wake of Star Wars or thereabout - i.e., the late 1970s. Part of this is the impeccably accurate portrayal of modern spaceflight, technology, et al, and part because of the gorgeous quality of the cinematography and special effects, which rival Star Wars and make it appear as though it were made in the late 70s.
Out of the Ghetto: Kubrick made this film specifically to bring science-fiction into the mainstream. He was fascinated by concepts of the genre but disappointed by most science-fiction books and movies. Drawing from external references (modernist literature, painting literature and philosophy), he deliberately approached the genre in a more realistic and enigmatic fashion. His film eschewed some of the genre trappings of World Building (space jargon, technology, alien species) and focused on how mysterious and bewildering space travel and alien contact could actually be. The groundbreaking special effects and greater sophistication made many people treat 2001 as an art-movie and Epic Movie spectacle rather than the usual B-Movie contempt which science-fiction was usually treated with.
Seinfeld Is Unfunny: Some viewers may find that the "Star Gate" sequence reminds them of the audio visualizers in MP3 player software.
Hell, even the computers look better than most of what came between this and the CGI era, or even the real life computers from The '80s.
And the technique used to create the "Beyond the Infinite" sequence — a camera trick known as "slit-scan" — was impressive enough to be reused well into the early CGI era. It was later used for ABC's "This is the place to be" ads of the early 1970s, as well as the Whooshing Credits for Superman: The Movie (which improved on 2001's techniques by using a computer-controlled camera) and a whole bunch of pre-CGI motion graphics work in The '80s.
The floating pen deserves to be mentioned as it is awesome in its simplicity; they simply used scotch tape to tape the pen to a sheet of glass, then rotated the glass around.
"Weird Al" Effect: As time progresses, it becomes more likely that the first time somebody will see something related to the film will be as a Shout-Out made in another more current work rather than in the movie itself.