The appliance used to connect computers and other devices to form a local area network (LAN) is called a router or switch. These devices facilitate communication between connected devices, allowing users to share resources such as files and printers, and access the Internet. Routers direct data packets between networks, while switches connect devices within a single network, optimizing traffic flow. By enabling connectivity and resource sharing, routers and switches are essential components for establishing efficient and functional LANs in both home and business environments.