What is a Storage Area Network, and which benefits does it have over different storage solutions?

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http://serverfault.com – I'm proposing this to be a canonical question about enterprise-level Storage Area Networks. What is a Storage Area Network (SAN), and how does it work? How is it different from a Network Attached Storage (NAS)? What are the use cases compared to direct-attached storage (DAS)? In which way is it better or worse? Why is it so expensive? Should I (or my company) use one? (HowTos)