Awesome Art: The poster art for most of the films are amazingly done (and pretty trippy).
Badass Decay: Freddy never joined the good guys (who would want him?) and always was a serious threat to them, but as the series went on, he became quite the jokester, and became goofier and more playful with people's dream sequences than he was in the first few films. This came to a head in Freddy's Dead, where he turned someone's dream into a Nintendo game, where you can tell he's having a great time.
Complete Monster: In the original continuity, Freddy Krueger is the dream-invading ghost of a child murderer, seeking to continue his spree. In life, he kept missing children's pictures of his victims in a scrapbook and posed as an unassuming, caring father before his wife discovered the truth; he then murders his wife with no hesitation. When he got off on a technicality, the people of Springwood lynched Freddy, at which point he makes a bargain with demonic forces to become, in his own words, "what nightmares are made of". He uses his newfound powers to find increasingly creative ways to torture and murder his dozens of victims for his own amusement by using their primal fears against them, all the while cracking sick jokes; afterwards, he harvests the souls of his victims and keeps them in a hellish state within his body to draw their power. By the time of The Final Nightmare, Freddy's body count had the hundreds, reducing Springwood to a child's graveyard and driving the adults into madness. He goes on to taunt a girl by appearing as her sexually abusive father, and even attempts to murder his own daughter Kathryn upon reuniting with her. Freddy makes it clear to Kathryn of his intent to continue his killing spree across the world until he runs out of victims, declaring "every town has an elm street." In Freddy vs. Jason, Freddy resurrects Jason Voorhees from Hell and loosing him onto Springwood to spread fear in his name. Growing annoyed that Jason is killing those he wanted for himself, Freddy makes an attempt to kill his pawn; it's revealed he murdered Lori Campbell's mother and attempts to rape Lori herself in her nightmare when she tries to help Jason. Freddy is one of the few Slasher Movie killers with a distinct and talkative personality, and has always simply been a sadistic monster who extends his victims' suffering for its own sake.
He's just as bad in the remake. See that page for more details.
Creepy Awesome: Freddy of course. He is a serial killer, but one with the ability to manipulate dreams to advantage and a ton of dark humor to boot.
Believe it or not, Freddy has an extraordinarily high amount of fangirls. The fandom likes to portray him as a woobie whose terrible past and upbringing were responsible for his psychosis, and that his love for his daughter was genuine enough to push him off the deep end when she was taken away from him. The fact that Robert Englund has said on multiple occasions that Freddy represents neglect doesn't help this image either. Even the lead actresses have admitted that Freddy has a seductive charm due to Englund's portrayal, albeit in a molester sense.
Love to Hate: Again, Freddy. He lacks any redeeming qualities but his skill set and dark humor make him extremely memorable.
Misaimed Fandom: It is truly disturbing how many 12 year old girls on DeviantArt draw cutesy anime-esque fan art about Freddy Krueger falling in love with their Distaff Counterpart OC of him they made. Who knew skinless demonically powered child rapists are so "cute"?
Misaimed Marketing: Though the marketing of his R-rated movies to said 12 year olds doesn't help. This was particularly prevalent with the heavy promotion leading up to the fourth movie, which downplayed his more gruesome aspects for mainstream appeal as a comic villain. This included an "interview" on MTV, his own rap number, and even a cheesy pop album (because "happy feet dancing to the beat" is apparently fitting music for a serial child killer).
Moral Event Horizon: Freddy Krueger crosses the line when he murders his own wife for discovering he was the Springwood Slasher.
Narm Charm: Freddy's make up isn't exactly a realistic depiction of what a burnt person would look like, but the look has become so iconic fans love it anyway. To the point that when the remake tried to go for a more realistic make-up, it was criticized as "generic".
Nausea Fuel: Freddy just loves invoking this, in the most ghoulishly creative of ways.
Nice Job Breaking It, Hero!: While you honestly can't blame the parents of Springwood for burning Freddy to death, this act of vigilantism ultimately allowed Freddy to become a powerful dream demon who would kill hundreds and by the end of the series, poses a global threat.
The whole concept becomes significantly less scary when you remember that Freddy only kills in one suburb (a fictional one at that), though it does get worse in Freddy's Dead, in which he states that "every town has an Elm Street" and makes clear that he's planning to expand his reach to the whole world.
As the original series goes on, Freddy's burned skin goes from horrifying to looking like a mild skin condition.
Physical combat isn't Freddy's forte, as whenever he tries to go up-close-and-personal with his victims, he loses. Freddy vs. Jasonaverts this by making him able to have an even fight with Jason.
Paranoia Fuel: The premise alone qualifies, obviously. It's one thing when you fall off a cliff in your dreams and wake up safe at home seconds later. But it's completely different when that can cross over into the real world.
Periphery Demographic: Despite Freddy getting most of the attention, and the slasher genre being typically marketed towards teenage and young adult men, the series has a notable feminist following due to its strong female protagonists who teenage girls easily relate to. This includes their lead actresses who encourage this demographic.
Sequelitis: Outside of primarily Dream Warriors and New Nightmare, every one of the sequels receives a sizeable amount of dislike. It didn't help that they were rushed so they could come out every year or two (Nightmare 4's script had to be written in 7 days due to the writers' strike).
Amanda Krueger. She was just a nun who wanted to help the mentally ill, but ended being captured and repeatedly raped by them, eventually giving birth to one of the worst known killers in existence. And even in undeath, she has to deal with her son's crimes.
Alice Johnson. Has all her friends and her brother killed by Freddy the first time out. Convinced she had him beat, he instead comes back and kills her boyfriend/father of her child, then he sets his sights on corrupting and possessing her unborn child? Life is not good to her. What's worse, all those deaths happen as an indirect result of her own dream power.
Freddy's victims in general tend to be portrayed more sympathetically than usual for the genre, featuring troubled teenagers whose problems are only exacerbated by the adults refusing to acknowledge Freddy's threat. This is especially evident in Dream Warriors in which the main crew are stuck in a mental institution due to being accused of self-harm.