Awesome, Dear Boy: Most actors just jumped at the opportunity of being in a Star Wars movie.
Beam Me Up, Scotty!: A minor one, but Anakin is sometimes associated with the line "This is so wizard!" or variations of the line; while the line is in the film, it's actually said to Anakin by Kitster.
Nathan Hamill (Mark Hamill's son) has an uncredited cameo role as a Naboo royal guard.
Cast the Expert: Darth Maul was played by Ray Park, who was a martial arts expert and stunt man, not an actor, although he has had a number of acting roles since then. This is one of the reasons that he had virtually no lines, and what lines he had were dubbed by Peter Serafinowicz.
Creator Backlash: Jake Lloyd isn't fond of his role in the movie, thanks to having been mocked about it since childhood. He went through a phase where he destroyed all of his Star Wars memorabilia and refused to talk about it. However, an interview with a Star Wars fan site has confirmed that he's mellowed out since then (it's still a sore spot, just not as raw anymore) and contrary to popular belief, he does not hate Star Wars after all the bullying.
Deleted Scene: This movie, like the others, has its fair share of deleted scenes. One of the more notable ones is where Qui-Gon and Anakin start to depart from Tatooine. One of Maul's sith droids attempts to tail them, only for Qui-Gon to suddenly slice it down, and then explaining they need to hurry back to the ship. This scene explained why immediately in the next scene, they were running back to the ship when the previous scene had them walking calmly from the market.
Another notable deleted scene, taking place after the podrace, involves Anakin beating the crap out of one of the other kids because he accused Anakin of cheating. Would have foreshadowed Anakin's prone to anger. Anakin's victim? A young Greedo.
Another scene had the pod the Jedi use to get to Theed surface in the city... and then have to be quickly abandoned before it goes over a waterfall. It was most likely cut for making no sense whatsoever.
Easter Egg: Aliens from E.T.'s species make a brief, split-second appearance in the Galactic Summit scene.
Fans critical of Jake Lloyd's performance as Anakin sometimes call him "Mannequin Skywalker".
Ric Olié is given the nickname of Captain Obvious, by virtue that nearly all of his lines are blatantly obvious assessments of what he sees.
History Repeats: Unlike the original trilogy, it wasn't a Troubled Production. Except that again George Lucas managed to get rain in Tunisia, in a Sahara region where it rarely occurs...
Old Shame: For Jake Lloyd, who soon after retired from acting owing to the mockery he endured in his youth as a result of his performance. Some friends of his from high school noted that he would sometimes fly into a psychotic rage when asked about it.
Also for Terrence Stamp, who reportedly disliked working on the film.
Recycled Script: The heroes visiting an underwater city to parlay with its ruler was originally a big plot point in Lucas' first draft for the original film, The Adventures of Starkiller: The Star Wars.
Sequel Gap: 16 years between this and Jedi's release. The lead actress was two the year the last one came out.
Spoiled by the Merchandise: The soundtrack to Star Wars Episode I The Phantom Menace was released before the movie itself came out. Anybody who picked it up got treated to tracks called "Qui-Gon's Noble End" and "The High Council Meeting and Qui-Gon's Funeral." Oops.
Star-Making Role: Both Ewan McGregor and Natalie Portman were well respected actors but were in either small roles or independent films. This movie made them big names. Ray Park also fits in, as his presence in the final battle was one of the most memorable parts of the movie.
On the other hand, it was a Star-Derailing Role for Jake Lloyd. He was never a star, but he had a respectable career as a child actor before this film. He did one movie after it and then quit acting. He said in a recent interview though that he's pursuing a career in film editing, so he isn't totally gone from the movie-making scene. One can only hope he has/had access to Wil Wheaton as a mentor.
In the earliest drafts, there was no Qui-Gon, Obi-Wan had the lead role, and Jar Jar had larger and more important role than being comic relief. Also when the crew returned to Naboo, the Trade Federation blockade was still supposed to be in place, and Anakin was going to land the Queen's ship by using his Force-intuition to divine the last possible moment to bring the ship out of lightspeed, before they crashed into the planet surface.
After Lucas decided to show the old Master-Padawan setup of the Jedi properly, he considered making Qui-Gon the Padawan.
George Lucas originally wanted Tupac Shakur to play Mace Windu, but he was killed before filming even began.