How do computer networks work?

Every computer on a network needs to have a unique identifying number, which is the IP (Internet Protocol) address. Actually, the IP address isn't the address of the computer itself, but rather the address of the computer's network interface.

A router is a device that connects computers to each other. As such, it has lots of interfaces (sockets) into which you can plug a network cable. The outside internet connection will have an IP address allocated to it by the ISP, and all the internal network connections will have their IP addresses allocated to them by the router. Certain ranges of IP addresses are reserved for internal networks, and one such range starts with 192.168 - so your outside IP address may be something like 1.2.3.4, your router's internal IP address may be something like 192.168.0.1, and your router will allocate IP addresses to the devices on your network, maybe 192.168.0.2 for the PC, 192.168.0.3 for your phone, 192.168.0.4 for your XBox and so on.

The router's main job is to "route" traffic around the network. It knows if a request for a web page has come from your PC it'll route the reply back to your PC. If the request for something else (game data for example) has come from the XBox, the reply goes to the XBox.

"Port Forwarding" is the process of telling your router to send specific traffic to a particular device on your network. So if you're running a game server on your PC on port 23456, you need to tell the router to send all data on that port to your PC and not to any of the other devices.

LAN just stands for Local Area Network, which is the sort of setup I've described above. The opposite is WAN (Wide Area Network) which kinda works the same way but some of the devices will be further away from the router (in an office building on the other side of town, for example).