A modem supplies your internet connection to your home.
Your pc can be connected by cable to the modem and access the internet.
A router connects to the modem and provides extra ports for more pcs to connect to the internet at the same time.
More usually, the router sends the internet signal wirelessly to pcs around the home so they can connect by wifi - without being connected by cable to neither modem nor router.
It is quite common for a modem / router to be combined in a single unit.
Also a router has an operating system and can be configured to provide security for your home network, whereas a modem basically converts analogue signals to digital and vice versa, hence the name MOdulator/DEModulator.
These days the differences are minimal .... "router" tends to be used to describe a device which contains
modem, router, dhcp, dns, nat, firewall wireless hub, hub/switch etc etc
so the configuration/application options are quite large.
a modem on the other hand - tends just to be a modem ......
in a "real" networking environment
a router supervises the data flow in and between subnets
(this is an over simplification ... but not a total lie ;-)