The internet is not one device or facility that can be attacked in order to cease the web from functioning. The internet is a network composed of many computers. Every website needs two things in order for it to be a part of the internet. A website needs a DNS (Domain Name System) address and a host. When you type in the URL for a website the DNS company translates that human-legible address to computer-legible address that directs the user to the host.
The host is the server that contains the actual website. It contains all the web pages and databases needed to operate and maintain the website. It also deals with any server-side scripts such as PHP. In most cases website rent servers from hosting companies because it is usually cheaper and easier to rent a server than to set up and maintain your own.
Data is sent to your computer in the form of packets. These packets contain small amounts of data that are combined together to create the web page or file you have requested. These packets do not all follow each other in a linear line on the same path. They can go from there start location to finish location in seconds, taking one of millions of possible routes. This makes the internet a very flexible and reliable marvel of human innovation.
However large chunks of the internet can be temporarily or indefinitely taken down from the internet. If the DNS company fails to direct the user to the host servers then the website will not be accessible. If the DNS directs the user to the host but the servers are not active then the website will not be accessible.
In 2016 a large scale Distributed Denial of Service Attack on the DNS company Dyn resulted in many popular websites such as Netflix, PayPal, Spotify, Twitter, Amazon, Reddit and many more websites being temporarily removed from the internet. The attack began at 7:00 AM EDT but was quickly resolved by the Dyn team at 9:20 AM. However, a second attack began at 11:52 AM and a third at 4:00 PM.
The hacking group responsible for coordinating the attacks never fully claimed responsibility so it is difficult to pin the attack on any one group. The group employed Internet-of-things devices with relatively low levels of security.
The internet was largely disrupted in the USA and parts of Europe.
It's important to understand that the internet can not be taken offline at the push of a button. In order for most of the internet to come down, it would require the attackers to attack many DNS companies, not just one in particular.