When you have a home router, this allows you to have multiple computer (or games consoles, phones, etc) connected and using the Internet simultaneously. Each device will have a unique private IP, which is normally assigned automatically by the router. These private addresses frequently start with 192.168....
The router uses a feature called Network Address Translation to map all outgoing packets so that they use one single IP address on the Internet. This is your public IP address, which is assigned by your ISP. The really clever bit about NAT is when a response arrives from a remote server, the router will automatically map that response back on to the correct private IP address for the correct computer. For most home users, the public IP is dynamic, meaning that it may change to other addresses that the ISP has been assigned for allocation to its customers.
Normally, when you are in a gaming situation, there are certain steps you may need to take:
1) Whatever device you are using for gaming may have to be on a fixed address on your private network. You need to find an address that is not already in use, is on the same sub-net as the router, and is outside the address range that the router can use to allocate addresses to other devices automatically.
2) You may need to set up port forwarding in the router so that packets arriving from the Internet that are not is response to an outgoing packet are directed to the correct device on your private network.
3) If you are hosting a server, you may need a Dynamic DNS service such as dyn.com or no-ip.com to provide a mapping from a fixed name for your Internet connection on to your public IP address, which may be dynamic and therefore changing.
I hope this helps you decide what you need.