Alright, so you are familiar with
rails s -b IP_ADDRESS -p PORT
To run the server on the localhost is simple (The default binding IP is 0.0.0.0)
To make it accessible from another computer on the LAN, you will need to specify your IP address and the port (normally port 80). If your LAN firework blocks port 80, you will need to specify another port then, such as 3000 or 8080.