I'm not sure if I understood everything you said, but here are my thoughts:
You need Xbox Live for Internet network games, or multiple Xboxes for LAN network games. I don't know any Xbox game that is compatible with a PC version of the same game; maybe they exist, but I don't have one. Team Fortress 2 PC vs Xbox does not work on a LAN, for example.
The Xbox itself can create an Internet connection through a modem, but you need to configure her accordingly in System/Network settings/additional settings. Since your PC obviously is configured to use dial-up connections, you can use the same settings that your PC uses.
The problem with that setup is that only one device can use the modem, since you always have to switch the plugs, and neither the Xbox nor the PC will ever be able to "see" each other (and use for example the Xbox as a Media Extender for the PC).
If you want both devices to talk to each other, you have to purchase a router. And while you're buying the router, make sure that you can connect the modem to it so that the router can dial up the Internet on demand. So both devices can go online at the same time.
Update: Of course, all of this ONLY applies when you were talking about a DSL modem, and not an old fashioned phone modem...
You need Xbox Live for Internet network games, or multiple Xboxes for LAN network games. I don't know any Xbox game that is compatible with a PC version of the same game; maybe they exist, but I don't have one. Team Fortress 2 PC vs Xbox does not work on a LAN, for example.
The Xbox itself can create an Internet connection through a modem, but you need to configure her accordingly in System/Network settings/additional settings. Since your PC obviously is configured to use dial-up connections, you can use the same settings that your PC uses.
The problem with that setup is that only one device can use the modem, since you always have to switch the plugs, and neither the Xbox nor the PC will ever be able to "see" each other (and use for example the Xbox as a Media Extender for the PC).
If you want both devices to talk to each other, you have to purchase a router. And while you're buying the router, make sure that you can connect the modem to it so that the router can dial up the Internet on demand. So both devices can go online at the same time.
Update: Of course, all of this ONLY applies when you were talking about a DSL modem, and not an old fashioned phone modem...