These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
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.
Better on DVD: Because you can watch the episodes in the right order, and without them being preempted by baseball games.
Complete Monster: Despite his initial appearance as an avuncular old man, Adelai Niska is quickly revealed to be the most vicious character in the series. A sadist with a love of personally conducting Cold-Blooded Torture on his victims, Niska is 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, 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. After commissioning the crew to steal important cargo from an Alliance train, the crew is horrified to realize that what they've stolen is the medicine needed to treat a city full of sick settlers. In his next appearance, Niska is seen carving up another failed employee, before getting his hands on Mal and Wash, whom he also 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 spends the remainder of the episode 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.
Fan Preferred Pairing: 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 a Base Breaker 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.
"I'm a leaf on the wind. Watch how I soar."
Ho Yay: 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.
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."
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.
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 ep.
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.
Murphy in "Shindig". Could also qualify from his snarky lines as a Cool Old Guy.
"Forgive me. I cannot abide useless people."
Monty, Mal's old war buddy, fellow smuggler, and fellow victim of Saffron.
"I shaved my beard for you, devil-woman!!"
Saved by the Fans: Resurrected for full length movie by a dedicated fanbase. Sadly averted, however, when it came to reviving the series as a whole. An attempt was made to buy the rights, but it failed without support from Joss Whedon or Nathan Fillion. Alan Tudyk noted that part of the problem was simply that the set piece for the ship is really expensive, so it's difficult to convince anyone to take it on.
Selfish Evil: Discussed Trope. When Saffron heads the plot to steal the Lassiter in Trash, half the crew basically ask "Why is the selfish, conniving bitch who tried to kill us, who betrays everybody for her own personal gain here?" The other half have expressions that say they're thinking the same thing.
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.
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 the big bad government.
The premise did come from journals of Confederate soldiers in the frontier. Whedon wanted to make a series based on the losers of a Civil War.
A show that presents the best and worst of both extremes isn't necessarily being apolitical. (See Accidental Aesop, above.)
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.