Simple answer: LAN IP is like an internal (local) IP address that's just for your home, work, school PC's & devices in your local network. The LAN IP won't work outside of the local network (LAN IP address for everything connects to a router on your LAN: modem, other computers, phones, printers and any other devices).
WAN IP is like your address for the web that your Internet Service Provider (ISP) gives you (gives your router and you).
- LAN IP: 192.168.1.1 -like an internal address only for your network, say at home or work). You can have different devices on your LAN that have addresses like 192.168.1.5 or 192.168.1.13
- WAN IP: 216.119.143.xx -an address the world sees for you when you browse the web.
- Gateway IP is the WAN address dedicated to the modem (like 22.214.171.124).
Note: A Virtual Private Network (VPN) puts you on its private network so no one sees your external IP that your ISP gave you, which means it's harder to track you. Make sense?
Static LAN means that instead of letting the router's DHCP assign IP address to a PC or device, you set at IP that is static, or remains the same--even if you disconnect from the LAN. Your ISP doesn't give you a static IP for your LAN, only WAN (if you need that). People use a static LAN so they can remember their Local IP address to make it easier for networking sharing, and for like shared printers or media players. You don't want DHCP to assign a new local IP address each time you connect or reconnect to your local router.