A modem is a device which converts a signal from one type to another, such as your cable signal, DSL, or phone signal to a standard ethernet signal used by your computer.
A router is a device which takes the ethernet signal from one source and acts as a traffic cop to direct it to its destination. In the context of a home network, it allows you to connect multiple computers to a single internet connection and there by share it. It also serves as a NAT Firewall, preventing systems outside of your local -home- network from entering and doing damage uninvited. For this reason, you should have a router even if you only have 1 computer. Routers can be purchased for less than $20 or upwards of several thousand if you need something better.