If you install a router that handles the Internet connections of your local network, it uses the public IP address which your Internet service provider assigned to you to communicate with other machines on the Internet. For reasons of simplicity and scarcity of IPv4 addresses, your ISP only assigns one address to your router, independent of how many machines are operating in your LAN. Each of your machines has a "local" IP address. Now the router has to employ a method to multiplex one "public" IP address to multiple "private" IP addresses in your LAN. This is done by NAT port forwarding.
All connections on the Internet can be identified by four parameters:
When one of the computers in your LAN initiates a connection to a server on the public Internet, it selects a source port number and establishes a connection to a certain destination IP address and port number. The source IP address for packets on this connection is the local ("private") address. However, this IP address is only valid in the LAN behind the router, but not on the Internet (hence the attribute "private").
So packets from the Internet can get back to your computer, the router maps your computer's local IP address and port number to a public IP address and port number. Typically, the port number is kept, but the local IP address must be replaced by the public IP address. The router modifies all packets sent to the remote server as long as the connection is alive. When the router receives a packet from the remote server for a mapped port, it uses the same port mapping to send the packet to the right computer on your LAN. For outgoing connections the router can establish this port mapping automatically.
Connections from 192.168.2.11:4242 to 18.104.22.168:80 are mapped to connections from 22.214.171.124:4242 to 126.96.36.199:80.
However, this automatic mapping is not possible for incoming connections. When someone from the Internet wants to establish a connection to one of the computers in your LAN, all your router sees is your public IP address as destination IP address and some port number, but it has no way to know to which local computer the packet should be forwarded. Therefore, you need to configure this mapping manually.
Usually the destination port number of incoming connection requests determines the local destination. You can set up your router accordingly, so that a certain destination port is always forwarded to a specific computer. If a new packet arrives from the Internet, your router replaces the public IP address and port number with their local counterparts that you had configured and then forwards the packet to the right local computer.
So from the standpoint of the public Internet all your local computers are just one machine with one public IP address and several established connections.
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