Actor Allusion: Niles mentions filling Frasier's floppy red shoes which could be a reference to his character Sideshow Bob. Guess who voices Bob's brother in several episodes? Yep — it's a mirrored Actor Allusion.
The first episode to feature "Sideshow" Cecil Terwilliger was aptly named "Brother From Another Series". In one bit, Bart has covered Cecil's eyes to surprise him, and pulls the standard "Guess who?" gag, to which Cecil replies, "Maris?" The Simpsons blandly lampshades this with a Frasier-esque title card reading "Frasier is a Hit Show on the NBC Network." Taken even further in "Funeral for a Fiend", which introduces Bob and Cecil's father Robert, played by...John Mahoney.
Grammer in particular became one of the series' main directors, helming thirty-seven episodes, many of them consecutively. He began in Season 3 with the famous "Moon Dance" (which focused on Niles and Daphne more to give him more time behind the camera) and directed with increasing frequency over the course of the rest of the series.
Edited For the Hallmark Channel: The Hallmark Channel routinely speeds up the end credits, resulting in the little skits looking like Charlie Chaplin shorts, with Kelsey Grammer lucky if he manages to get up to the line "Scrambled eggs all over my face" by the time the Grub Street logo appears.
Fake American: John Mahoney was born and raised in Blackpool, Britain. He became a US citizen in 1959, but sometimes his old accent pops up.
Daphne's least favorite brother, Simon (played by Australian Anthony LaPaglia) has a terribly unconvincing accent. Mostly played for laughs, though.
See also any guest actor (not-British) playing Daphne's random boyfriends in earlier seasons. See British Accents above.
Jane Leeves is British but cannot do a Manchester accent for toffee. God help the woman if she ever needed directions to the Trafford Centre. Word of God is that they wanted Daphne's accent to be working class and understandable to Americans so Jane Leeves adopted that particular accent. 
Gil Chesterton is played by an American-born English actor and displays many symptoms of I Am Very British. In one of the first episodes, Frasier comments on his "phony accent."
Real-Life Relative: Kelsey Grammer's then-wife Camille (whom Kelsey divorced in February 2011) makes a cameo as a partygoer dressed as Eve in "Halloween". Frasier hits on her, but she leaves with another man... dressed as Satan no less.
Kelsey Grammer has said that his favourite episode was "The Show Where Diane Comes Back" (the plot of which was Diane Chambers of Cheers and Frasier Crane's eventual reconciliation), because it gave him the opportunity to finally reconcile with Shelley Long, the actress who portrayed Diane. Also-
Frasier: You have a wonderful career.
Nanny G: But nothing ever changes! Do you have any idea what it's like to play the same character for twenty years?
Niles and Daphne have a son named David in the series finale, named after producer David Angell who died in the 9/11 attacks.
Diane Chambers twice in dream sequences, once in a season 3 makeup episode. Shelley Long and Kelsey Grammer had an on-set rivalry on Cheers, but they made up in the course of filming the episode.
In "Head Game", Niles becomes the shrink to a famous basketball player. This was interesting as the plot was meant for Frasier, but as Kelsey Grammer was being treated for his alcoholism he was unable to fulfill the role.
Stunt Casting: Parodied. The people who call in to Frasier's radio show are usually famous celebrities, but you'd never know unless you tune in to the closing credits.
Technology Marches On: In the early seasons, there are frequent references to pagers, and Niles is the only one of the cast wealthy (and pretentious) enough to have a cellular phone (his first one isn't quite a brick, but you can watch cell phone technology change with his upgrades). One episode even highlights how relatively rare the devices were when Frasier notes that a recently-arrived professional juggler must have been contacted on her "car phone", prompting Niles' near slack-jawed shock that "Street performers have car phones?!" Of course, most of the various Fawlty Towers Plot styled antics wouldn't have worked quite the same if the characters could just call each other at any time.
A seventh season episode has Roz enthused by the fact that Cafe Nervosa has put in a phone line to allow people with (rather clunky) laptops to go online.
Kirstie Alley is the only actor from Cheers to not reprise her role on Frasier, reportedly because Scientology does not believe in psychiatry, and Frasier is a psychiatrist. In the later episodes of Cheers, around the time Lilith and Frasier were going through their separation, Frasier and Rebecca nearly ended up in bed together. Wasting a perfectly good UST plot...
Kirstie Alley claimed in one interview that her being a Scientologist had nothing to do with her never appearing on Frasier, and at least one of the show's writers (Ken Levine) said in a radio interview that they never considered having Kirstie Alley on, mainly because they weren't sure how to write her in, as Rebecca and Frasier hardly interacted on Cheers. Alley also said once she auditioned to play one of Frasier's girlfriends of the week, but was turned down, because the producers thought that would be kind of weird.
Lisa Kudrow was originally cast as Roz, however after the first few days of filming the producers decided that her quirky humor didn't fit the part and they hired Peri Gilpin instead (according to Kudrow, Gilpin was their first choice, but switched her for Kudrow for some reason, then simply restored order by bringing Gilpin back.) Plus, imagine how this would have affected Friends if Kudrow had remained on Frasier.
Frasier's brother Niles wasn't in the original concept and hadn't been mentioned on Cheers. The inspiration for the character came after the producers saw a headshot of David Hyde Pierce and noted his brotherly resemblance to Kelsey Grammer.