Your router is assigned a dynamic IP address by the Internet Service Provider. You can find out what it is by Googling "What is my IP address". The router assigns a local IP address to your computer. Your computer may have two local IP addresses, one for Wi-Fi and one for Ethernet. You can find out the local IP addresses by looking at System Preferences/Network.
To access your WeatherCat server via the Internet, you need the dynamic IP address and you need to enable port-forwarding to the computer hosting the WeatherCat server. Port-forwarding allows requests coming in from WeatherCat/iPhone via the Internet to access your WeatherCat server. Port-forwarding is a kind of "It's okay for this outside guy to access something on my computer without asking permission." thing. Domain name services like DYN help by assigning a name to the IP address and, since the dynamic IP address may change, update the name with the current IP address.
Some WeatherCat users have static IP addresses. They can use the IP address directly without fear of having the address change. You can use your dynamic IP address with the caveat that you may need to change the address you use in WeatherCat/iPhone occasionally.