There are two "IP addresses" you have to concern yourself with, assuming that you want your device to be "visible" from computers outside your LAN, out there on the internet.
a) The "local" address... the address of your server/ whathaveyou, your Arduino-connected-to-the-LAN. That will usually start 192.168.... and must be set to something nothing else on your LAN is using. That is your device's address, and that's all you need to "talk" with it across the LAN.
If you're sending or receiving "stuff" beyond the LAN, then the "stuff" carries a "return address"... the address of your LAN's connection to "the internet", as seen from outside your LAN. If your router has a "static IP address"... big company, university, etc... then a number will do. Home users rarely have static IP addresses. (And will know from their (high) bill, if they do.) They need to use a domain name and DYNDNS.
All explained more carefully at Arduserver.com