Here's the brief synopsis (by Jonathan Lasser from the Contemporary Literature pages of About.com) :
"Fat Charlie" Nancy is perpetually embarrassed by his father, a dapper old man who talks to everyone, loves karaoke, and plays practical jokes on everyone-including Fat Charlie. After the elder Mr. Nancy's funeral, Charlie discovers two things: his father was Anansi, the story-telling trickster god; and Charlie was not an only child.
Fat Charlie soon meets his brother, Spider, who is everything that Charlie wishes he was: confident, well-liked, and the center of attention. Spider, who inherited their father's powers, immediately moves into Charlie's apartment and begins putting the moves on his fiancée, Rosie. Fat Charlie, desperate to rid himself of his troublesome brother, enlists some unorthodox assistance, and things progress from there.
This is how Neil Gaiman described it, when he was sending out proof copies to booksellers and reviewers:
My new novel is a scary, funny sort of story, which isn't exactly a thriller, and isn't really horror, and doesn't quite qualify as a ghost story (although it has at least one ghost in it), or a romantic comedy (although there are several romances in there, and it's certainly a comedy, except for the scary bits). If you have to classify it, it's probably a magical - horror - thriller - ghost - romantic - comedy - family - epic, although that leaves out the detective bits and much of the food.
You can see a very short video (just over 90 seconds) of the author explaining the plot of the book (including a lime. Not the lime from the book, a stunt lime.)
If you have a choice, I highly recommend listening to the audio version (CD or eAudio) of this. Neil Gaiman wrote Anansi Boys with his friend Lenny Henry in mind. Lenny Henry reads the audio book. It's marvelous.