"Vivian, you are so naive. You'd believe that boy if he told you he was a rap star whose latest album just went platinum!" (cue Aside Glance from Janet Hubert-Whitten rather than Will).
In response to a character's truthful claim, Will sarcastically snaps, "Yeah, and I just won a Grammy".
In the pilot, Hilary and Carlton's establishing moment is a fight in which they argue over the correct pronunciation of "Porsche". Alfonso Ribiero was in Silver Spoons, wherein his character's father and Ricky Schroeder's character's father had a very similar argument.
Any time Carlton dances, it's hilariously dorky unless it's to Michael Jackson, in which case he busts out moves worthy of Jackson himself; Ribiero was a backup dancer for Jackson before he was cast on The Fresh Prince.
Cloud Cuckoolander Judge Robertson, played by Sherman Hemsley, gets in a heated argument with Philip once he learns that Philip's running against him for Superior Court Justice, culminating with Robertson shouting "Lionel, show him to the door!" before walking up the Banks' stairs. Hemsley showing up later in the run to reprise his role as George Jefferson makes this Hilarious in Hindsight.
The Danza: Subverted. Will Smith plays Will Smith. However, in real life, Will's actual name is Willard, while in-show, Will's name is William. On an episode of Late Show with David Letterman, and again on The Graham Norton Show, Smith said that Alfonso Ribero advised him to do this, because people would be calling him by his character's name for the rest of his life, as demonstrated by the fact they went out and were called "Will!" and "Carlton!"
Will's best friend "Jazz" was played by DJ Jazzy Jeff.
Executive Meddling: At the start of Season 4, Janet Hubert-Whitten was replaced by Daphne Maxwell Reid. No reasons were given to the public for over a decade, but Hubert-Whitten fills that Will Smith played a role in her sacking.
Alfonso Ribeiro confirms that the whole cast was like one big family, except for Hubert-Whitten, who was difficult to work with and toxic to be around, and really killed the mood on set.
This was further corroborated by James Avery in a June 2013 edition of OMG! Insider, in which he remarks that he constantly had to remind Hubert-Whitten that the show was The Fresh Prince of Bel-Air, not "Philip and Vivian of Bel-Air."
I Am Not Spock: Defied: Will Smith chose to call his character "Will Smith" because he knew he would be referred to as his character's name for the rest of his life. According to interviews, he was acting on the advice of Alfonso Ribeiro who hasn't quite escaped this. "When we go out together, people are like: 'Will, Will! Carlton!'"
Irony as She Is Cast: Carlton's actor, Alfonso Ribeiro, was, and still is, an accomplished dancer. The character, largely, can't dance to save his life.
The Other Darrin: The role of Vivian Banks was recast after Janet Hubert-Whitten violated her contract when she became pregnant; the pregnancy and birth of her character's child, Nicky Banks, was worked into the plot to accommodate the situation. After she was fired, Hubert-Whitten's character was played by Daphne Maxwell Reid for the remainder of the show's run. According to both Will Smith and Hubert-Whitten, they had difficulties working together long before her pregnancy. The other cast members also admitted to having difficulties working with Hubert-Whitten mainly due to her being toxic to be around and for trying to make the show be more about Aunt Vivian and Uncle Phil, even though Will was the central character.
A small case. Will's mom in the opening sequence is the "mom figure" used in Smith's DJ Jazzy Jeff and Fresh Prince videos (particularly "Parents Just Don't Understand").
Frank Schaeffer was played by Dietrich Bader in season 2, and by Robert Torti in season 3.
Post Script Season: The third, fourth and fifth season finales were written to be a respectable farewell in case they didn't get another season. The sixth season Grand Finale was written with the full knowledge it would not continue.
A literal case with the fifth and sixth seasons; the show was actually cancelled during the fourth. See Un-Canceled.
Technology Marches On: In "The Baby Comes Out", Vivian uses a big cordless phone with an even bigger antenna to beep Will.
At one point during the end of "Papa's Got a Brand New Excuse", Will goes off script from his rant about the present he got for his father to a speech based on his own life experiences and how he will be a better man than his father (eventually breaking down in tears). You can tell by the confused look on James Avery's face when the change occurs.
In "The Script Formerly Known As", Will's over-the-top lip-synching "performance" of "And I'm Telling You I'm Not Going" was not scripted, but a gag made up by Will Smith himself.
Tom Hanks Syndrome: Will Smith ran into a career block when trying to keep up the hip, street-savvy black guy he started portraying with this show. Ironically, no one remembered him as "The Fresh Prince" after Independence Day.
Un-Canceled: The show was actually cancelled during its fourth season, with "The Philadelphia Story" serving as the Grand Finale. The overwhelming response, with viewers writing in by the truckload to NBC and Will Smith, convinced the network to go back on this decision, allowing it to run for what became two more seasons.