Basic Networking: Pity the Fools That Don't Know It.
Local Area Network
This includes all the devices behind the router. It's typically on a subnet with addresses that look like this:
192.168.x.x
10.10.x.x
Wide Area Network
This is the World Wide Web and endpoints connected to it. IPs (Internet Protocol) can be virtually anything that is not one of the ones reserved for local network.
Domain Name Servers
The phonebook for the Internet. All domain names resolve to an IP, and that IP allows it to communicate across the web.
There are root level DNS servers, and then there are DNS servers at your Internet Service Provider and even local office, and IP resolution is cached all the way down.
Your Domain Name Registrar may ask you to wait 72 hours to see the changes when you make a transfer to another DNS server. So if you are doing this operation or others, make sure you have it right before the change.
Network Address Translation
The Router changes the IP address hit from the outside world into the IP used on the LAN:
The only things exposed are those ports that are forwarded from the router.
Ports are the sockets through which the communication occurs: