This will cover the TV-verse for the Dresden Files, not the book-verse. The show was based on the series of books by Jim Butcher, but the canon of the TV series is not exactly the same as the book canon.
BASIC PREMISE
Harry Dresden is a wizard, who works as a sort of supernatural PI in Chicago, helping his clients resolve problems related to the magical world
Putting "Wizard" on your door is like putting up a sign that says, "Lunatics Welcome." - Harry Dresden
CHARACTERS
Harry Dresden:
Perpetually broke wizard do-gooder. Harry is the son of a sorceress and a stage magician. He represents the last of the Morningways, a family of some power and notoriety in the magical world. Harry's mother died when he was 3; the cause of her death was never known. Harry's father died when Harry was 11; the cause of his death was thought to be a heart attack, but was later revealed to be murder.
After his father's death, Harry went to live with his uncle, Justin Morningway. As an adult, Harry learned that it was Justin who murdered his father. Harry and Justin had an altercation, which resulted in Harry accidentally killing Justin with black magic. Harry inherited a great deal of money from his uncle, but he gave it all away. He remains wealthy on paper due to still owning the estate home of his family, but he does not live in it.
Harry is a bit of a loner, having no family and few friends in the world. Harry possesses magic, but it takes effort to cast spells and his technique is sometimes described as unrefined. When he gets frustrated, Harry's powers are sometimes shown to short out electricity. Because of this, he is often seen in his home using candles instead of electric lights. He drives an old style jeep (presumably, no on board computer to fry) and does not own devices such as a cell phone, computer, or television.
Harry: I dropped out of high school, remember?
Bob: And you wonder why your life is an unending series of insurmountable intellectual, financial, and emotional hurdles.
Bob:
He serves as a mentor and friend to Harry. Where Harry lives in a state of scruffy chaos, Bob has poise, taste, and elegance. He is also a snarky, perverted know-it-all, which makes him a lot of fun.
Bob appears as a human being, but is a ghost with no corporeal substance and is sometimes seen passing through walls (or even through Harry, which Harry dislikes intently saying it gives him the "heebie-jeebies"). He is bound to his skull and cannot move far beyond the its vicinity. Effectively, whoever owns Bob's skull also "owns" Bob. He spends most of his time at Harry's place since he cannot travel anywhere unless Harry takes the skull with him. Bob's skull used to belong to Harry's uncle, and Bob first met Harry when Harry moved in with his uncle after his father died. It was Bob who trained Harry in magic.
Bob was once a powerful wizard known as Hrothbert of Bainbridge, but was condemned to his ghostly existence for the crimes he committed in life. There is not a lot of detail as to what exactly the crimes were, but it had to do with his repeated efforts to resurrect his lover from the dead.
Although Bob is extremely physically limited by his imprisonment, he still possesses a wealth of magical knowledge, which he uses to help Harry. Bob is sometimes seen "writing" spells in the air (like Bob himself, these writings have no substance; they disappear when anyone passes through them). Bob can also take readings from the physical world and can take on the appearance of someone other than himself.
Bob: ... the rent is due next week.
Harry: Bob, you're a spirit. You live in a skull. You have no physical needs. You're really going to bitch about the rent?
Connie Murphy:
A Chicago police detective who often consults Harry on more bizarre cases. She thinks the whole "wizard" thing is an eccentricity, but deeper than that she is very troubled at the thought that magic could be real so does not acknowledge the possibility that Harry is what he says he is. She is divorced with a daughter, and had a strained relationship with her father, which seemed to be on the mend.
Murphy and Harry have developed a friendship, but the relationship is sometimes complicated when Harry falls under suspicion for crimes. Harry once had to save her by using black magic, which also complicated their relationship since it it caused her to later suffer emotional problems that she did not understand. In the end though, she appreciates the fact that he is her friend and that his work results in closed cases.
Murphy: Harry, I keep dreaming about you.
Harry: Good dreams, bad dreams, what you got?
Murphy: I don't know. Sometimes you're pretty scary in them. Other times not so much.
Morgan:
He works for the High Council, the governing body of the magical world as a warden, a sort of High Council law enforcement officer. He distrusts Harry considering him dangerous due to Harry's use of black magic in killing his uncle. However, Morgan and Harry sometimes work together to achieve common aims.
Morgan: Some things never change -- always blowing things up.
Harry: I am not *always* blowing things up.
Morgan: Right, sometimes you set things on fire.
Justin Morningway:
Harry's maternal uncle, who took 11 year-old Harry in after Harry's father died. Justin murdered Harry's father by crushing his heart with a Voodoo doll. As fate would have it, this is exactly how Harry killed Justin. Justin made a brief reappearance in life when a copy of himself (a sort of magical clone) restored Bob to corporeal being and recruited Bob to resurrect dead!Justin. Bob did the resurrection, but then killed Justin then died himself (rendered again into a spirit bound to his skull since the curse he was under was never lifted.)
Justin had ambitious plans for changing the magical world, but it is never revealed what those plans were, only that Harry was to have a role in the change.
Justin: Whatever I've done, I've done for love.
Harry: Love? You don't know what love is.
Ancient Mai:
She is a very prickly and powerful leader of the High Council, who does not like Harry. She commands respect and obedience. It is not clear exactly what she is, but it is suggested that she may be a dragon.
Mai: You've made me unhappy, Dresden. Bad for me, worse for you.
Bianca:
A vampire of some power, who gave shelter to Harry when the High Council was hunting for him after he killed his uncle . Harry, at the depths of despair at that time, wanted her to kill him. She refused because she wanted him to owe her a favor, which she later collected by having him help her find out who was setting her up for a downfall. She can be beautiful and charming on the exterior, but vicious and manipulative underneath. There are moments when she seems fond of Harry, but at the same time she recognizes it is in her nature to be a monster.
Bianca: Don't I frighten you?
Harry: You don't frighten me.
Bianca: That, my dear boy, is why one day I might just have to rip your throat out.
Malcolm Dresden:
Harry's father, a stage magician who possessed no real magical abilities. He and Harry lived an itinerant life as Malcolm would work different jobs. Malcolm was very close to his son and protective of him. He refused to give Justin custody of Harry, which ultimately resulted in Justin murdering Malcolm. He died in front of Harry when Harry was 11.
Malcolm: Harry, I want you to listen to me, okay. What you can do, what you have, that is a loaded gun and there are people who are going to want that gun and they are going to want to use you to do things that you're not going to want to do.
Margaret Dresden (nee Morningway):
Harry's mother, this character is never seen, but is referred to. She possessed powerful magical abilities, but something managed to kill her when Harry was 3 years old. Who or what caused her death is not known. Justin, her brother, described her as "starry-eyed", "impetuous", and full of "dangerous ideas", but it is not clear what those ideas were. Harry wears a protective bracelet that once belonged to her.
Mai (to Harry): You're just like your mother, full of ideas about how the world should behave.
Sid Kirmani:
Murphy's colleague. Not much is known abut Kirmani, but he is shown to be loyal to Murphy and skeptical of Harry.
Kirmani (to Harry): Let me guess, a couple of summers ago you were a Jedi knight and now you’re a wizard named Harry? It’s good marketing I guess but couldn’t you come up with something a little more original?
FANDOM:
The show only lasted one season on the Sci Fi Channel, but the fandom is a dedicated and creative bunch.
Here on LJ, we have:
dresdenfilestv for general discussion of the show
dresdenfic for Dresden Files fan fiction
dresdenflashfic for periodic fiction challenges
dresdenfiles100 for drabble challenges
skull_boy_love for Harry/Bob shippers
harry_murphy for Harry/Murphy shippers
dresden_art for icons, wallpapers, screencaps, etc.
Meanwhile, off of LJ we have:
Sci Fi Channel for the official site of the show
The Archive for Dresden Files fan fiction
DF on fanfiction.net for fan fiction (note, all DF fic is in the book section even stories based on TV canon)
Jim Butcher Forums - TV Series for discussion of the show hosted on the book author's site
Sci Fi Channel Dresden Files Forum for discussion of the show
Dresden City for fan efforts to help get the show renewed