Lets break down the term WAN. It means WIDE AREA NETWORK. It isn't similar to a cable modem. Cable Modems use a large cable as a physical connection that gets processed in the device as a WAN, much like an ethernet connection would, but on the network of the Internet provider.
Now lets go further. Any network is a set of 2 or more connected devices. With only 2 devices connected directly, it's easy to know where to send information back and forth, only one place to go to or get it from. When you have an unknown number of potential connections and an unknown type of information, you need to find ways of separating and sorting everything, then a way to queue everything in proper order, and a way to send back a response of some sort to prevent or bypass errors. Routers do this by assigning each physical ROUTE a number, and then different sets of data also get numbers called ports. The numbers are IP addresses, sets of 8bit numbers that represent binary ones and zeroes, that enables extremely large, though not infinite, networks to be set up. WAN is a type of network for your outside internet connection. It doesn't separate anything, it connects the router to the internet on one route. However, that route's address is controlled by the Service Provider's networking, and very similarly to the way your INTERNAL NETWORK is set up, even if the actual signal comes in over different wiring (it's still a physical route). WAN separates nothing, it connects the router to the internet. There is programming in the router that accepts messages from one of the ip addresses in your internal network, then checks a local list of addresses for a match, and when it doesn't find one, it sends the message out over the WAN to an OUTSIDE DNS that will check other DNS lists and send the message to the destination on the internet--presumably to the computer hosting the website. There is something called NAT or NETWORK ADDRESS TRANSLATION that can run which allows your router to HIDE your computer's address from the internet, and send the message as if it came from the router itself, which in a way, does isolate you from internet dangers, however, it still advertises port numbers and other data that firewalls are now scrambling as a defense mechanism. DNS is DOMAIN NAME SERVER. A machine that keeps a list of machines and how to reach them is a DOMAIN NAME SERVER. Without one of these, a network is only internal and will be very slow, as it has to attempt to check every route for matching data before it can actually send the message through. In the programming, networks can actually be separated from one another by being MASKED. Because BINARY is all 1 or 0, machines can see one another when the addresses and the mask pattern align properly. Using this, you can set up multiple networks in the same physical area, using the same dumb (no OS) switches, provided the devices all get their address and subnet manually set. However, if you have a smart switch, you can set up some physical ports for each network you want to run, and they will pass data through.
DHCP is the Dynamic Host\Client Protocol. It's programming that will create the numbering system and send it to every machine connected to it. Running this on a router is common now in homes, but running it on routers in small businesses is also common. Larger corporate networks are usually handled by several servers, one or more for DHCP--which means there could be multiple networks running simultaneously, self contained, and somewhat fault tolerant--one or more DNS--each network needs some way to know the fastest route to each connected device--and one or more GATEWAYS--which are just the connection to the internet. GATEWAYS usually run NAT, but don't have to. Most Corporate networks have a firewall as well. This can be a large server type that acts like a router, or built in mathematics on the GATEWAY\Modem, or any combination. Stronger firewalls are more costly.
WAN DOESN'T separate you from anything. It's just on a different LOGICAL or mathematical network than your internal network.
Much like this, some internal mesh networks actually use a similar networking style called STATIC ROUTES, built into the routing table or DNS lists, that allow separated networks to access one another much more securely. By seeing the master network as a WAN or outer network, they allow data to pass only to specific services or devices on other networks, but they cannot see each other in any other way, and are otherwise isolated from one another. Most Operating systems have their own DNS list called a HOSTS file, which can allow you to specify this pathway, but most routers will have a method of adding these kinds pathways so that they can be "CASCADED", allowing multiple NAT and firewall impelmentations for heightened security. This also allows some networks to use something called VPN or VIRTUAL PRIVATE NETWORKING. By setting up two of these servers in different geographical locations, setting one up to contain the internal DNS and routing of the internal network and allowing it to pass out a few addresses outside of the other set handed down by the main internal network (on the same masking but outside of the normal set handed out by the first DHCP), then setting the other up to send a message to that router on a specific port number to be forwarded directly through to the VPN server behind it, one VPN server is the HOST, the other the CLIENT or SATELITE, and the two offices can share connected systems and resources.
By combining all of this, network technicians build vastly complicated networks that allow fault tolerance, and continuing function so corporate assets maintain value. By doing the same, hobbyist Home users can hide their presence on the internet with VPN services, and even run several different networks for home security and other fun things that all run on the internet of things (which is just physical things that are connected to the internet), while keeping each network separate and maintaining the speed and usefulness of their main computer networks.