What purpose do modems and routers serve in a network environment?

Recently, I completed a networking course and was confident in my abilities.
Modems weren’t mentioned in this course and I didn’t know they existed. It was my belief that routers were responsible for all internet communication. This would explain how simple a network structure would look.
Internet -> Router-> Switch -> Hosting
My research revealed that modems are actually the devices that connect to the internet. Routers, however, only route within a local network. This leaves me very confused, as I had assumed that that was the job for the switch.