SANs usually use iSCSI or Fibre Channel protocols to connect storage to workstations. Sometimes called targets and initiators respectively.
A layer of software is put on top of the SAN to manage traffic between the storage and workstation. Think of it like a traffic light, but for your data.
Most SANs use a 2nd network, often called a metadata network, to separate out high performance traffic and metadata, which is basically all of the permissions.
This SAN type design allows for higher performance as it puts the high performance traffic on its own dedicated connection. A SAN is typically more complete and more expensive and used for very specific needs
A NAS is a much more common type of network design. In fact you can turn any Mac, Windows or Linux workstation into a NAS by enabling file sharing on your storage.
NAS systems use native file sharing protocols like AFP, SMB and NFS to share storage from a single user or server, to many.
NAS systems use ethernet, which is typically 1GbE or 10GbE to connect users.
NAS systems are generally less expensive and more simple. Performance will depend on how the network is set up and what hardware you have.