All the address on the User side of your Router are from one of three ranges that are called "private" IP address. They are used only for local networking, and are NEVER directly visible on the Internet at large.
When you use your Router, it 'acts as your agent' on the Internet, and substitutes if own, ISP-Provided IP address for the local address, and sends your requests on their way. this is called "Network Address Translation', and it means your computers, printers and server are, by definition, NOT reachable from the Internet.
By default, any unsolicited packet sent to your Router is Discarded, unless you requested a response from the Internet. To request that packets of a certain description be forwarded to a particular private IP address on your Home network, you need to enable 'Port Forwarding' in your Router.
Port forwarding allows packets to be sent to your computer, but it does not provide a stable Domain-name for that connection. If you are using this for commercial purposes, you should BUY a Domain-name from an Internet service provider. Iff doing e-commerce, you will also need to BUY a certificate for HttpS: use.
if you are just using this for your own purposes, there are sites that can provide the ability to add the current IP address your ISP is using for your Router to its server and a user-supplied Domain-name so that your port-forwarded device can be found. But if your ISP changes your IP address for any reason, you will have to reset the address. if you choose to 'grab' a domain-meme and a legitimate company Buys it, you lose that battle -- the guys who legitimately buys it wins.
Allowing others to send packets to your device is MUCH more dangerous that an ordinary connection, and you should password and/or encrypt every connection. You WILL see frequent requests to log in with root access coming from all over the world. On the Mac that is receiving these packets, turning on its FireWall should be considered, but is not the ONLY thing you need to be doing to protect that Mac.
If all you are doing is multi-player gaming, many such games can enable the ports they need once you tell your Router Port that forwarding is allowed. You may not need an Internet-visible domain-name just for gaming.
This will take some research on your part.
