A router is a device that connects your wireless or wired devices to a modem or a fiber ONT. The internet connection in your home is shared between the access point and your personal devices. This sharing creates a local network where a router acts as a hub and protective firewall that allow control of network settings, security, and what devices can access the network.
It is not required that the modem and router be separate devices. Many providers offer wireless gateways or hybrids that are both modems and routers. Most routers today come with wireless capability (Wi-Fi).
Your provider will assign an Internet Protocol (IP) address to the router, not the modem.