What are the Differences between a Modem, a Router, a Gateway and a Modem Router?

Modem
A modem is a small box that connects your household to the Internet using cables. It acts as a digital translator, taking an information signal (Internet data) from your cable or phone lines and making it accessible to your computer.
Router
A router is works with your modem to join networks together and allow multiple devices like phones, tablets and computers to use a single network through a wired or wireless connection. The router’s main function is to create and send out the Internet Wi-Fi signal in your home.
Gateway and Modem Router
Modems and routers can be combined into a single box, and gateway is another term for a combined modem router. The primary benefit to using a modem router or gateway is the simplicity of only having a single device to set up.
How They Differ
In the traditional home network chain of command, the router talks to the modem which talks to the internet service provider, resulting in an internet connection and Wi-Fi. A modem will give local Internet access to a single device, but requires a router to connect multiple devices via Wi-Fi.
When your home network uses a modem-router combination device or a gateway, it talks directly to the Internet Service Provider to connect you to the internet and enable a Wi-Fi connection. Since gateways and modem routers combine the functionality of a modem and a router, only one device is required to achieve the same result.