The local IP is what you use for port forwarding -- this says to your router "for incoming traffic on the specified ports, send it to the computer at 192.168.0.3".
For hosting a GE:S server, you don't actually need the public IP to be used in any settings. The public IP is what others will use to connect to your server. They make a connection using your public IP, which goes to your router. Your router then looks at your forwarded ports and sees that connections to port 27015 go to 192.168.0.3, and so it then sends that traffic to your computer.
Your local IP will likely change from time to time -- if it does, your port forwarding rules need to be updated to reflect that. You can see
this guide for how to set a static IP for your computer. The steps are the same for Windows 8 & 10 as well -- the only difference being how you can get there. In 7 and later, you can always just right-click the network icon in the taskbar and choose "Open Network and Sharing Center", which will take you to Step 7.
Your public IP will likely also change from time to time, but you don't need to set this anywhere for hosting to work -- it only changes what IP your friends use to connect to your server. In a game like GE:S, this is no huge deal since there's only ~20 servers that populate the list, so it'll be easy to find your server. In a game like TF2 however, with its thousands of servers, you'd want to just give your friends your IP so they can connect directly. You usually can't do anything to keep the same public IP, short of paying your Internet service provider for a static IP.
