Internet depends on 13 Servers.

Internet, as we know it today, depends on 13 DNS name-servers.
They are managed by the US coverment or are managed by organisations around the world.
Today NTIA announced (controlled by the US goverment) that it stops controlling those nameservers.
At the beginning of 2015, all DNS root servers will be managed independently by international organisations. A good move.

Here is the list: http://www.iana.org/domains/root/servers

There are many of them but they all depend on 13 root DNS servers.
Psychically, every one of them consists of a cluster of servers, but virtually, there are only 13 of them.

To answering the question: "Can you shut down the whole internet with a terrorist attack?", the answer is no.
First, you can discover the physical places via the IP addresses of those 13 servers quite easy, but you don't know what the infrastructure is behind the address.
It can be placed all over the world.
Second, everything is cached by slave servers. The number will be way bigger.
3rd, you will only try to attack/forget the names, not the infrastructure.

In other words: Your action will be just as stupid as IS actions for example. Totally useless. Even if you own all the bombs available.

How does the DNS logic work behind this?

Enter "www.facebook.com" in your browser and if your computers DNS cache doesn't know www.facebook.com, it will ask it's own DNS server.

-Your PC asks it's own DNS server "I am searching for www.facebook.com", the DNS server answers (if it isn't cached there) "I have no idea, but I have a list of root servers, I will ask them" (as you can see in the picture, my computer already knows the 13 root servers).
- The DNS server asks to the root servers "I am searching for www.facebook.com" and the root server answers "No idea, but I know who manages .com, I will ask them".
-DNS server asks the .com nameserver "I am searching for www.facebook.com" and the .com nameserver answers "No idea but i know who manages facebook.com",
-DNS server asks the facebook.com nameserver "I am searching for www.facebook.com and the facebook.com nameserver says "Yes, I know it", here is the IP address in my record you was looking for...".
-DNS server passes the IP address back you your PC and stores it in it's cache. Now your browser knows the IP address to receive the Facebook page.