amanoffewwords,
It all depends on the modem. I have run into modems which use a crossover cable when connected directly to the PC, others I have seen with straight patch cables. Then some come with crippled routers that only allow one computer to be connected to it, even if you use a router and spoof the MAC Address (Those can be reprogrammed), some which require the computer to have a specific name in order for the modem to work. Some need PPPoE, others do not, etc.
One of the more interesting modems I have seen is a DSL Modem from Lucent which has a built in Router. You do not use PPPoE on the computer, but you have to enter the DNS Server Addresses under Networking on each computer in order to get it to work.
There are many different varieties, without knowing the model of Modem it is all guesswork.
Take Care,
Richard
P.S. Aiglon with a router, you do not have to have any particular computer turned on in order for the other computers on the network to be able to access the Internet.