This is the first time you see Buffy's Dad in the show, he appears 4 times in Season 1 Episode 10 "Nightmare", Season 2 Episode 1 "When She Was Bad", Season 5 Episode 21 "The Weight of the World", and his last appearance is in Season 6 Episode 17 "Normal Again".
Nitpick: Aldo and Willow were attempting to preform the famous duet from Puccini's Madame Butterfly. Willow was dressed to play the title character Cio-Cio-San and Aldo was playing her new American husband Pinkerton. In the actual duet, Cio-Cio-San does not come in until after Pinkerton's first five lines (where Willow asked if it was her turn). The first prompting look Aldo gave her was factually inaccurate.
Goof: At the beginning of the episode, an exterior shot of Sunnydale High is shown for a few seconds. This shot was taken from "Teacher's Pet" (1x04). You can see Willow sitting on the bench wearing her red shirt and plaid pants from when she first met Ms. French in "Teacher's Pet." However in the next scene, which takes place only a few seconds later, Willow is wearing a black and brown shirt and a flowered skirt.
Willow still has a picture of herself and Giles in her locker.
When Wendell says that he doesn't hate spiders and that in fact he loves them (spiders hate him), Xander asks "that's platonic right?" After the praying-mantis-woman tried to mate with him in "Teacher's Pet" (1x04), Xander may not consider this a rhetorical question.
When Xander reaches down to pick up the candy bar (just before the clown attacks), it appears to be a Hershey Bar. But if you look closely, you can see that the name has been altered. Both H's are gone, and the R has been changed to a P. The name of the candy bar is now "EPS EY."
Nitpick: After Billy wakes up, everyone obviously still remembers what happened over the last few days. Just moments before he wakes up, the hospital staff is panicking: yet immediately after he wakes up, you can see the hospital staff wandering through the hallways, as though it were a perfectly normal day. Unless Buffy and the Scoobies only remembered because of their close proximity to Billy, that seems far too fast to get over such traumatic events.
Nitpick: Buffy's tombstone shows the dates "1981-1997." This birthdate contradicts both of the dates (1979 and 1980) from "I Robot, You Jane" (1x08). Ironically, this date (appearing, as it does, during a dream sequence) is more reliable than the ones mentioned previously, and is the one that seems to stick for the rest of the series. Also, this episode had to have taken place after Jan. 1st, 1997, because in the episode "Witch" (1x03) the sign in the gym says "1996 Cheerleading Tryouts."
Nitpick: It is unclear how the vampire Buffy made it to the hospital in daylight.
We learn that Giles can read five languages. On a normal day, anyway.
Goof: The Master's artificial thumbnail is visibly falling off in a few shots of the opening teaser.
Goof: In the beginning of the episode, when Buffy is talking to Willow before class, she closes her locker door. But seconds later, the camera cuts back to Buffy's open locker, and she closes it again.
Goof: In the climactic scene where Buffy is going up against "The Ugly Man," he has two "dead eyes" contacts on, but later he only has one. He has two when Buffy is about to leap on him in the hallway, and he has one when he is slumped over in the room.
Cordelia: (to Wendell) Hello? Doofus! You're in my light.
Xander: Wendell, what is wrong with you? Don't you know that she is the center of the universe, and the rest of us merely revolve around her?
Master: Fear is a wonderful thing. It is the most powerful force in the human world. Not love, not hate... Fear! We are defined by the things we fear. (goes to the large cross) This symbol, these two planks of wood, it confounds me. Suffuses me with mortal dread. But fear is in the mind. (puts his hand on the cross and holds on while it burns) Like pain. It can be controlled. If I can face my fear, it cannot master me.
Joyce: (to Buffy) You just have to remember that your father adores you. No more than I do, by the way.
Willow: I don't like spiders, okay? Their furry bodies, and their sticky webs, and what do they need all those legs for anyway? I'll tell you: for crawling across your face in the middle of the night. Ewww! How do they not ruffle you?
Xander: I'm sorry! I'm unruffled by spiders. Now, if a bunch of Nazis crawled all over my face...
Willow: (to Wendell) What do you mean, you love spiders?
Xander: It is platonic, right?
Giles: The boy has been in a coma for a week. How can this be possible?
Buffy: What, am I knowledge girl now? Explanations are your terrain.
Buffy: Could I be seeing Billy's asteroid body?
Giles: Astral body, and I don't know.
Buffy: This isn't real. You can't be free!
The Master: You still don't understand, do you? I am free because you fear it. Because you fear it, the world is crumbling. Your nightmares are made flesh.
Joyce: You want to go to school?
Buffy: Sure. Why not?
Joyce: Ok. Good day to buy that lottery ticket.
Xander: So our dreams are coming true?
Giles: Dreams? That would be a musical comedy version of this. Nightmares. Our nightmares are coming true.
Willow: So, why is this happening?
Xander: Well, that explanation was shorter than usual. It's Billy! Who's Billy?
Xander: It'll be faster if we split up to look for her.
Giles: Good idea.
Willow: Uh, faster, but not really safer.
Wendell: I don't know what to say about that.
Xander: There's nothing to say. You saw two hundred insects, you gonzoed. Anybody would have.
Wendell: They're not insects. They're arachnids.
Xander: They're from the Middle East?
The Master: So, you're the Slayer. You're prettier than the last one.
Xander: I'm not worried. If there's something bad out there we'll find, you'll slay, we'll party!
The Master: What's the fun in burying someone if they're already dead?
Willow: When Buffy was a vampire, you weren't still, like, attracted to her, were you?
Xander: Willow, how can you... I mean, that's really bent! She was... grotesque!
Willow: Still dug her, huh?
Xander: I'm sick, I need help.
Willow: Don't I know it.
Buffy: This is a dream.
The Master: A dream is a wish your heart makes. This is real life.
Buffy: Scary. But there are a lot scarier things out there than you... and I'm one of them.
Vieni La Sera from Madame Butterfly by Giacomo Puccini - The opera duet that Willow is supposed to perform
According to the first Watcher's Guide, Joss Whedon stated this is the episode where Xander "officially" grows up, due in large part to his punching out the clown.
It was Alyson Hannigan's idea for Willow to suffer from stage fright, which is how the writers decided to make performance anxiety her biggest nightmare.
Billy: I had the strangest dream. And you were in it, and you...
This line is very similar to what Dorothy says at the end of The Wizard of Oz, when she wakes up from her dream.
The Master: A dream is a wish your heart makes.
It's amusing (and slightly ironic) that The Master is familiar with Walt Disney. This is a line from a song in Disney's 1950 movie Cinderella.
Xander: Your balloon animals were pathetic. Everyone can make a giraffe!
Balloon animals refers to the practice of twisting long balloons into shapes representing animals. The giraffe is a particularly common balloon animal. Clowns often do balloon animal shapes.
Buffy: Could I be seeing Billy's asteroid body?
This refers to the theories of Theosophy, which teach seven different planes of existence with a corresponding body. The Astral body is able to leave the physical and move about the Astral plane.
Willow: Why is she so Evita like?
Both a play and movie, Evita is based on Eva Peron, wife of Argentinean president Juan Peron. Immensely popular with the people of Argentina, she is portrayed as self-centered and opportunistic.