The youngest Baudelaire is only a baby and only intelligible to her brother and sister (at least at the beginning). However, she is extremely intelligent, and in addition to having four very sharp teeth as a weapon, she also demonstrates admirable cooking skills later on.
And in later books, instead of gibberish, she often says words (or partial words) that relate to her response, or at least the topic being discussed. For example, when describing a sword fight, she says "Flynn", when somebody mentions a train, she says "Esoobac", when talking about going undercover, she says "Dragnet", and when somebody asks her to do something impossible, she exclaims "Unfeasi!"
Little Miss Badass: She once fought against a sword-wielding hypnotist with her teeth, and held her own for a good while.
The main villain of the series. His goal is to get the Baudelaire fortune, no matter where they go and how many stupid disguises he has to wear. He's revealed to have a connection to the shadowy organization known as VFD.
Abusive Parent: To the Baudelaires...an abusive foster parent, anyway.
Woobie, Destroyer of Worlds: Maybe. It's implied that Beatrice and/or Bertrand Baudelaire and/or Lemony Snicket killed his parents with poison darts during a performance of La Forza del Destino.
You Killed My Father: The film makes it more blatant that he was responsible for the death of Baudelaires' parents and the burning of their mansion than in the book, as Klaus finds the giant magnifying glass responsible for it and exposes it by burning the wedding contract.
Esmé Gigi Genevieve Squalor
One of the Baudelaires' many foster parents turns out to be evil and becomes Count Olaf's girlfriend. She's a wealthy woman ridiculously dedicated to keeping up with every ludicrously inane fad that comes about.
The mysterious narrator of the series who holds a torch for a deceased woman named Beatrice.
Alter Ego Acting: Daniel Handler and Lemony Snicket - separate characters in the books themselves.
Author Appeal: Mild example - Daniel Handler is something of a gourmand, and hence the Lemony Narrator never misses an opportunity to describe some delicious dish, even providing a salad recipe in the midst of an urgent-seeming message to his sister embedded in the tenth book.
Plot Based Photograph Obfuscation: Snicket never shows his face in photographs, but there are several possible explanations for why this is, and most such photographs are only seen by the audience in his author bio rather than by the characters.
This also applies in-universe. A note in the Quagmire diaries indicate that Snicket's face is never seen in a photograph. And indeed, when the Baudelaires find a photo of their parents, there is an unidentified man with his back turned next to them.
Stalker with a Crush: Inverted - Lemony's a good guy, but he does stalk the children of the woman he loved but couldn't have but should have had.
The mysterious initials of a shadowy organization that everyone - from Olaf to the Baudelaires' parents - is connected to.
The Omniscient Council of Vagueness: V.F.D., and specifically the transcript of the meeting of the vague "Building Committee" in the Unauthorized Autobiography - even the author didn't know some of what was being discussed here, and he was technically in attendance.
The Baudelaires' friends are a identical brother and sister whose brother Quigley died in a fire. Referring to themselves as "triplets" (just because Quigley's dead doesn't mean they were born twins), they help the Baudelaires out and get kidnapped for their trouble. Duncan is a journalist while Isadora is a poet specializing in couplets. Later on, Quigley is revealed to have survived.
Tyke Bomb: In her second appearance, Count Olaf and Esmé Squalor adopt Carmelita Spats as a Tyke Bomb, but she's so thoroughly spoilt by Esmé as to be utterly unhelpful, and after demanding lessons on how to spit in exchange for shooting someone with a harpoon she's ditched by Olaf; he later turns his attention to Sunny as a possible replacement.
Count Olaf's Troupe
Olaf has a large variety of henchmen he calls his "acting troupe."
Ambiguous Gender: The Person of Indeterminate Gender, a.k.a. the enormous person who looked like neither a man or a woman.