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 David Letterman and again on Graham Norton, 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 is no longer part of the cast, and the role of Vivian is recast to Daphne Maxwell Reid. No reasons were given to the public for over a decade, but 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 Whitten, who was difficult to work with and toxic to be around, and really killed the mood on set.
Further corroborated by James Avery in a June 2013 edition of OMG! Insider, in which he remarks that he constantly had to remind 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, 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 was and is an accomplished dancer. The character, largely, can't dance to save his life.
And in something of a subversion, Jada Pinkett (now Jada Pinkett Smith because she married Will) was turned down for a role as Will's girlfriend due to a lack of chemistry. Irony As She Is Not Cast, maybe?
Money, Dear Boy: Had it not been for Will Smith's financial troubles (he overspent, and didn't pay income taxes) during his initial music career, the series would never have happened.
Old Shame: In-show example. Geoffrey's secret was that he was a long distance runner who cheated while representing Britain at the 1976 Olympics. He got ahead of the other racers by taking a cab to the stadium. He was immediately found out and his gold medal was taken away. In England, he is "The Shame of a Nation."
The Other Darrin: Janet Hubert-Whitten was replaced by Daphne Maxwell Reid in 1993.
Lampshaded in one episode, where Jazz comments that there's been something different about Vivian ever since she gave birth to Nicky and Will then breaks the fourth wall to stare at the audience.
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 canceled during the fourth. See Un-Canceled.
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 from the look on James Avery's face (when he becomes confused) 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 canceled 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.