>>956252
Physically the simplest home network connecting with modem to the net looks like this:
computer - router - modem - internet
If you plug the computer and the router together they can communicate, you should be able to reach the router from your computer no matter the status of the modem. You have a LAN right there.
The modem just converts the data stream from digital to analogue and back - it needs to be on, but it doesn't play much role I think.
The router has an inner and an outer IP. The inner faces the LAN the outer faces the internet.
When you connect your computer and the router together your computer should get an IP from the router. This is called DHCP, since the router is a DHCP server too - it serves IPs to the connecting computers.
To use the internet you need an outer IP for the router. ISPs use DHCP servers to serve IPs to their customers machines - machines can be routers.
With the router you should be able to request an outer IP for it and the ISP will serve you one, depending on their settings for you (what you payed for) a dynamic or static IP.
They should have tables of static IPs on their DHCP servers.