Your modem will still provide your Internet, but the router will negotiate the connection to your modem - not your computer(s). The computers will connect to the router and the router will share your Internet between them.
He's saying you need a router. The router connects to your cable or dsl modem and your computers connect to the router. Things can get hairy after that depending on how you connect to your ISP, whether its PPPoE or some other way. Best to get that information from your ISP before you begin.
And you'll need both parties (computers) to be either connected to the router via the ethernet cable, or a wireless "card", if its not inbuilt to the pc (like a laptop would be)