Hopefully I understand you're question. If not, I apologize in advance. Your computer should have 1 IP address per network interface card (NIC), although they can have multiple. This means that your wireless card will have an IP address and any wired ethernet cards will have their own addresses. Your router will (typically) have two IP addresses: one is public and one is private. The public address will allow you to communicate with your ISP and therefore the internet. The private address is to communicate with a subnetwork (subnet) that will have all your personal devices: laptops, pc's, printers, etc. All your computers and other network devices will use the public address of your router to communicate outside the subnet, a.k.a. with the internet. In other words, all the traffic from the internet will go to the router (via the public address) and then the router will forward the traffic to the appropriate subnet device (via its private address and subnet addresses of the devices).
Hope this helps! Please send me a message or respond to this answer if you have further questions. :)