When you are within the network of the router, you use the IP of the Arduino.
Can you log into the router and see all the clients ?
It would be for example something like this:
192,168,1,200 : computer
192.168.1.201 : other computer
192.168.1.203 : arduino 1
192.168.1.204 : arduino 2
If you are using the computer (at 192.168.1.200) you type in the browser: http: //192.168.1.203
However, if you are outside you home network, for example on the internet, you have to open a port in the router. In that case you can use the IP of your internet connection and let the router translate a specific port to the right local IP number.
http://whatismyipaddress.com/
You have a router that is a client ?
It depends on how you have set it up. Does that router have a DHCP server ?
Is your primary network 192.168.0.xxx ? and your local network after the router is 192.168.1.xxx ?
Or is also your primary network also 192.168.1.xxx ?
The easiest way is if your router does not use its DHCP server and also your primary network is 192.168.1.xxx.
In that case your primary network and the local network are fully transparant and the primary router (the one 'above' your router) handles the IP numbers also for the Arduino.
I have a router after a router after a router myself. My second router makes its own network with a DHCP server enabled, and a computer on the first network can not reach my Arduino. I have to configure the second router to make it possible to pass it to reach my second network. I have wireless devices on my second network, but also wireless routers as clients. They don't use DHCP, they just act as pass through wireless clients. So an Arduino connected to the second network has the same IP number as when it is connected to a wireless client device.
It's getting confusing, can you make a drawing of your network, with the IP numbers ?