Your router will be given it's own IP address (for example 192.168.0.1) the other computer(s) will connect to, this is known as the default gateway. It will also have a subnet class (normally 255.255.255.0).
Each computer has it's own unique IP either generated off that (for example 192.168.0.102, 192.168.0.103, etc) or are manually added. If these aren't different you will have issues when both computers connect at the same time. The same subnet class is needed to join the network.
IP Classes / Default Subnet Mask: Class A: 1.0.0.0 to 127.255.255.255 / 255.0.0.0
Class B: 128.0.0.0 to 191.255.255.255 / 255.255.0.0
Class C: 192.0.0.0 to 223.255.255.255 / 255.255.255.0
Class D: 224.0.0.0 to 239.255.255.255
Class E: 240.0.0.0 to 255.255.255.255
Out of the five classes, the first three, A, B and C are used on the Internet by its users in order to communicate, while the rest, D and E, are reserved for other reasons. In most cases, you will always be working with Classes A to C. Class C is default for most routers.
The Router connects to the internet and gets it's own unique IP address for online access. Internet sites such as whatismyipaddress.com can automatically detect my IP address when accessed through the Web browser.
On a home network you may not find that computer's IP address but the address of a network router instead.
An IP address is a unique number that every computer connected to the internet is assigned.