What is the modem and what is router? Are they different?

A modem is different from a router.

The word Modem is the short name for Modulator-demodulator.
A modem is a device or program that enables a computer to transmit data over telephone or cable lines.
Computer information is stored digitally, whereas information transmitted over telephone lines is transmitted in the form of analog waves.
A modem converts between these two forms.

Broadband routers are designed for convenience in setting up home networks.
A broadband router supports file sharing, Internet connection sharing, and home LAN gaming.

Modem是調製器、解調器的合稱。 它是一種能夠使電腦通過電話線同其他電腦進行通信的設備，它所做的工作是把存儲在電腦中的資訊從二進位格式轉換為能夠以電話線傳輸的格式。 它把一個個字元轉換為聲音信號並傳到電話線上，而電話線另一端的Modem 則把聲音信號還原為位元（Bit）。

router(路由器) 大致上是用來連結兩個或以上的網路之用.