The authors use the great title metaphor to setup and explain the two types of organizations. Here, centralized organizations are like spiders, with a head and command and control system, and distributed organizations are like starfish, with no central brain and nearly-autonomous parts that can function so well indpendently that you can cut off a starfish's leg and the leg will grow an entirely new starfish.
Practical examples follow, with studies showing how Alcoholics Anonymous, Craigslist, Wikipedia, Napster, and Skype all flourished as they embraced distributed models or aspects.
The text does a good job elaborating the differences between centralized and decentralized organizations, and it also concedes that many organizations have evolved to use a hybrid model. Perhaps the best example of a hybrid organization is eBay, which is decentralized in how it lets users sell to each other and centralized in how it has established a reliable rating system and relationship with PayPal to ensure transaction security.
Bottom line: The Starfish and the Spider is interesting and and well worth reading.