A modem connects the source of your internet from your ISP and your home network, whether you use a cable provider, fiber optics from FIOS, satellite such as Direct TV, or a DSL or dial-up phone connection. The modem connects to the router —or directly to a computer — using an Ethernet cable. Modems are different for each type of service; they are not interchangeable.
A modem is a modulator/de-modulator. Data transmission on the Internet occurs via cables in analog format. But electronic devices like computers and phones only understand digital transmission. A modem is the translator between the public Internet and a computer network; the modem modulates and demodulates signals to convert analog signals to digital and vice versa. This lets devices communicate on the Internet.
The difference between a modem and a router is that a modem connects to the internet, while a router connects devices to Wi-Fi.
Until now we have talked about the typical home or small office network. There is another domain where you get Internet service and use a modem—your cell phone. Cell phones have a built-in modem that translates the analog signal from radio waves that are used to communicate with cell towers. This is also called a wireless modem, and should not be confused with the integrated modem/wireless router used in home networks
Features of Modem are as follows:
Modem works on Data link layer (Layer 2 of OSI Model).
It Facilitates a connection to the internet by transmitting and receiving data over telephone lines.
It provides one for connection to ISP. Next one to connect with router(For large networks).
It transfers data in the form of Packets.
It is an Inter-networking device.
It Can connect to one PC using Ethernet port.
It is vital part of internet connection.
It does not have any data security measures as such.
A modem can work independently without a router, delivering information (such as Internet access) to a single PC.
The specific techniques used to encode the digital bits into analog signals are called modulation protocols.
Source:google and multiple sources