How exactly River's psychic abilities work is up to interpretation. The two most common theories are that either she's precognitive and can sense things before they happen, or that she possesses exceptional telepathic awareness of her surroundings which, coupled with her extreme intellect, lets her predict what's going to happen through analysis. Or maybe both at the same time.
River herself. Some interpretations, especially post-Miranda, have her as a generally stable person who is coping with her trauma and abilities. Others assert that she is completely insane and needs constant supervision with brief bouts of lucidity, while some would argue that she's just a girl who is struggling to cope with her own trauma and brain damage and wavers wildly between madness and lucidity.
Saffron goes through this both in- and out-of-universe. The crew is uncertain as to whether or not she's an absolute sociopath, or if she's got some redeeming qualities. Mal, at least, seems to be convinced that at least part of her breakdown in "Trash" was genuine.
Was Mr. Universe just an eccentric who married his own Sex Bot? Or was he Properly Paranoid, knowing the kind of enemies his line of work would bring? And considering his fate in The Movie, it could be argued that having a Sex Bot wife proved to be the right move.
Anti-Climax Boss: In the post-Serenity comic Serenity: Leaves on the Wind, Jubal Early is revealed to have survived his initial encounter with the crew and marks his reintroduction by effortlessly beating up the soldiers guarding the Alliance Operatives he's meeting with and then slaughtering Bea's entire crew and blowing up their ship. He once again infiltrates Serenity and manages to subdue the crew one-by-one... then he gets blindsided by Kaylee and knocked out in one hit. He's then dumped out of the ship with the rest of the trash.
Better on DVD: Because you can watch the episodes in the right order, and without them being preempted by baseball games, and see three episodes that were never even aired in the first place.
Adelai Niska is a Soft-Spoken Sadist proud of his feared reputation amongst the criminal community, and is obsessed with ensuring the horror stories surrounding him are all true. When he first meets the crew of Serenity in "The Train Job", Niska shows off the mutilated corpse of his wife's nephew to solidify his reputation in their eyes and to show them what price they'll pay should they fail him in stealing a shipment of medicine intended for sick settlers. In his next appearance, in "War Stories," Niska carves up another failed employee, before getting his hands on Mal and Wash, whom he proceeds to torture for hours. Eventually Zoe, Mal's first mate and Wash's wife, offers to buy Niska's captives off of him, but Niska tells her that with the money she has, she can only afford one of them and tries to force her into a Sadistic Choice. When Zoe ruins his fun by immediately picking her husband, he responds by saying there is enough money for some of Captain Reynolds. He then cuts off Mal's ear and gives it to her. Niska then starts torturing Mal to death, only to use advanced technology to bring him back to life so Niska can have the pleasure of torturing Mal to death for days.
The Blue Gloves/"Hands of Blue", from "The Train Job", "Ariel", and the Serenity: Those Left Behind comic, are a duo of independent contractors hired by the Alliance to capture the Tam siblings. The Blue Gloves hunt the Serenity across the 'Verse, killing anyone in their way as they go, and employ a variety of schemes to capture the two, notably allying with Dobson in an attempt to kill the entire crew. The most evil aspect of them, however, is their use of a sonic device that painfully kills their targets by rupturing their insides until they melt, which they use to kill an entire Alliance building, systematically checking all of the bodies and murdering any who still cling to life to ensure none survive the massacre, just because a couple of the guards had interacted with the Tams, and the Blue Gloves seek to keep all information possible to themselves.
Jubal Early debuts in "Objects in Space" as an aloof yet sadistic Bounty Hunter seeking to turn in the Tam siblings for the reward on their heads. Incapacitating the crew of the Serenity and threatening to rape and torture Kaylee to keep some in line, Early is seemingly beaten after he is revealed to be a Psycho for Hire who only has his job because he enjoys hurting others. Returning in the comic book Serenity: Leaves on the Wind, Early kills an entire staff of security guards just for prohibiting him entry to a meeting, and later bombs a smuggler ship in another attempt to capture the crew of the Serenity, killing a dozen innocent people in the process. Despite his claims that he follows a code and is only doing his job for the money, Early is actually nothing but a cruel bully who views killing people as a "passion".
From a scene where Malcolm wins a duel with a truly obnoxious and stuck-up opponent that had it coming:
Malcolm: Sure. It would be humiliating. Having to lie there while the better man refuses to spill your blood. Mercy is the mark of a great man. (lightly stabs him with sword) Guess I'm just a good man. (stabs him again) Well, I guess I'm all right...
Fan-Preferred Couple: There are three main, widely accepted pairings for this show: Mal/Inara, Simon/Kaylee, and River/Jayne. Simon/Kaylee, however, while running on UST for the entire series, was made canon at the end of the movie, and Mal/Inara probably would have produced some sort of result anyway if the show had gone on longer - so River/Jayne, which is only hinted at in the show, is the only real example of this trope portrayed here. The latter is also notable because it is something of base breaking among the fans, but it is also the only consistently popular fan-preferred pairing.
Foe Yay: Mal and Saffron have heaps of this, as do the Tam siblings with Jayne.
"Funny Aneurysm" Moment: From "Shindig", when Zoe and Wash are Talking in Bed about what to do if Jayne decides to take over the ship. Just try laughing at Wash joking about how to properly eulogize Zoe after watching her at his funeral in the Big Damn Movie.
Wash: I'm a leaf on the wind. Watch how I soar.
In that same vein, there's Wash's fantasy about being rich in the "Better Days" comics that take place in-between the show and the Big Damn Movie. In it, he's piloting a luxury cruiser big enough to "land a planet on" and takes a break to play with his child. Thanks to his death in Serenity, only a small part of that comes to pass in later comics: The one-shot comic "Float Out" (which, ironically, was collected in the same volume as "Better Days") reveals that Zoe became pregnant before his death, and during the events of "Leaves on the Wind", she gives birth to their daughter, Emma.
Genius Bonus: The opening line of the song is "Take my love / Take my land" and goes on to say that the singer doesn't mind because he's still free. Machiavelli's The Prince says the two things you can't take from people without inciting revolt are their women (love) and land.
According to Adam Baldwin, there would have been deliberate Ho Yay between Simon and Jayne if the series had not been cancelled. Sean Maher also stated in a convention interview that he believed Simon had a crush on Jayne, something that would likely have been used in a future storyline.
Mal also refers to Simon's mouth as "pretty" at least once in the show and once in the movie.
There's also noticeable Les Yay between Kaylee and Inara and Kaylee and River, with the latter especially noticeable in the pilot and "Objects In Space."
I Am Not Shazam: Firefly is the name of the class of ship, not the ship itself. Serenity is the name of the ship, the first(ish) episode and The Movie.
And as Inara points out, it's not The Serenity, it's just Serenity.
Iron Woobie: Mal Reynolds and the full weight of The Chains of Commanding. He lost his entire homeworld to war, lost most of the people he ever cared for, then lost the war. After that he embarked down a life of crime that chafes at his principles, and alternates between Rage Against the Heavens and looking after his crew, often at the expense of his health.
Jerkass Woobie: Jayne, at least in the moments where he shows a softer side, which almost always only happens when something terrible has happened to him.
Like You Would Really Do It: In the pilot, everything is set up to make the audience think Kaylee has died, up to and including a main character outright stating it. Even those familiar with Joss Whedon's Anyone Can DieSignature Style probably didn't really think he'd kill off a top billed character in the first episode.
Jayne comes very close to crossing it in "Ariel" when he tries to sell Simon and River to the Feds. He comes within seconds of getting Thrown Out the Airlock, but his regret and shame afterwards show that he is not completely irredeemable.
Niska crosses it when he kidnaps Mal and Wash and starts torturing them to death. Bad move.
Alternately, the point where Zoe jumps the gun on his Sadistic Choice and he responds by giving her "small refund".
"Heart of Gold". Rance. Willing to take out the entire brothel just to get to his kid, and is willing to be sexually serviced in public, because he thinks that's what women are for. Rather telling for someone who doesn't even survive the episode.
FOX will always be the network that killed this show. To the point that the only reason Dollhouse got a second season was them not wanting the fans burning them in effigy.
Netflix for their 2013 April Fool's joke in which they claimed Firefly was going to continue as an original series on their service. To say that many fans were not pleased for getting their hopes up is a bit of an understatement.
No Yay: To some fans, River seems too severely damaged (mentally and emotionally) to be a mutually consenting adult, so efforts to ship her tend to run into Unfortunate Implications. She's doing better after the events of Serenity, but the extent of the improvement is unclear.
Stoic Woobie: Simon has to deal with pretty much everything his sister is going through, and received little to no support from either of his parents when he made his move to get her out of there. This is also Zoe after the movie, and depending on how badly the war affected her, possibly before and during the series as well.
Unfortunate Implications: One of the criticisms of the series was the fact that despite taking place in a future where China was a massive superpower and Chinese influence could be felt in everything from dialogue to clothing, there were almost no actual Chinese (or indeed any Asian) characters in the show. Even the Tam Siblings, who had a Chinese surname, were played by white actors.
What an Idiot!: The guy who just beat you up and defeated your gang of mooks offers to give you back all the money owed to your boss and let you go, no hard feelings. Do you A) gratefully accept the unexpected mercy, or B) threaten to hunt down and murder the man while standing between him and his rotating engine intake?
What Do You Mean, It's Not Political?: People seem to want to read politics into Firefly everywhere, with critics interpreting it as everything from an anti-Big-Government libertarian parable, to a racist, pro-Confederate perspective on the Wild West. The show is, in fact, fairly apolitical; with the exception of some possible feminist overtones (which are typical of Joss Whedon's work), Firefly does not seem to have one specific political "message" as much as people would like to believe. Whedon even qualifies that, if anti-government messages do sneak in, that is because the story is essentially Mal's story, and it all comes from his perspective as a bitter ex-soldier who fought against a government that he personally considered evil.
The Woobie: River. Between her madness, the fear the others show around her, and her traumatic backstory (plus Summer Glau's excellent acting) she gets tremendous sympathy from the fandom. Kaylee also gets similar treatment thanks to the fact that every villain seems to be laser-guided to target and hurt the sweetest person on the whole ship. Wash has some Sad Clown tendencies. Book doesn't really know where he belongs in life, and what with an abusive childhood never really had a family. Inara may also qualify, given her secret.