Are all the computers getting their IP addresses using DHCP? If so, make sure that the DHCP server (or whatever is giving out IP address info) is also giving out the DNS nameservers and domain. Otherwise, you won't be able to connect to a website using its webname URL (http://mywebsite.com, for example).
Also, are you using a router with a switch (or one built into it)? If not, then one of the computers has to act as the router by enabling Internet Sharing.