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Internet Mail
In order to run your own domestic mail server with an address of you@domain.com you first need to register domain.com at a registrar. I use GoDaddy and Name.com, but please look around to find a registrar to fit your needs and budget - it usually is about $10,- a year.
Second you need to set up the DNS at your registrar in the sense that you tell their system to make an A-record for the name server.example.com pointing to the public IP of your modem.
Third you need to set up an MX record with a preference of - say - 10 and the host would be the A-record you've just created. Please follow the instructions of the registrar you choose, the'll have a manual or you can contact their customer support.
Fouth you need the set up port forwarding within your the device that gets the public ip (usually your modem) for ports 25 (TCP), 587 (TCP) to forward the ports to the internal ip address of your server.
You can usually consult the manual of your modem's manufacteren online.
Background information
DNS
Domain Naming System is a way of exchanging a name like apple.com for an ip address like
17.149.160.49. It's like one big conversion table, which you need because computers don't know what to do with domain names except forward the request you put in to a DNS Server that you have to specify somewhere, a popular one is Google's (8.8.8.8 and 8.8.4.4) which you are free to use. You don't really *need* a local DNS server.
Private IPs
The ip's of 192.168.*.*, 172.16.*.* and 10.0.*.* are typical Local Area Network addresses.
This is all within your house or building.
Public IPs
Your modem is assigned the public IP address normally, unless you set up your modem in bridge mode, in which case the router would need to handle authentication (PPPoE).