Port forwarding is used when you have a router and you wish to let users connect to your server through it. if you wish to host your server for local reasons, it is not required that you do so. Keep in mind that port forwarding might cause security risks.
When port forwarding, it varies on how your router will ask you for the information. If you don't understand on how your router want's you to input the information, try visiting PortForward.com for a tutorial. If you still need help, please visit the Minecraft Forums and create a thread asking for help. You need to register
Once you have managed to locate your router's admin page, and find the Port Forwarding page; hit add new service (may not work) or custom service. When you get a page asking to setup the new rule, it should prompt you on what you want to call it. You may name it as you wish but for simplicity, name it "minecraft". Then you want to look for "type". Select TCP. For the ports (internal and external), enter 25565. If it asks for anything else other than output IP (or internal IP, server IP), leave it alone and continue.
For the out put IP address, use the following steps.
Windows
Locate to your desktop and hit start. From there, either search for cmd, or on previous versions of windows press "run" and type in cmd and hit enter. Once you have completed those steps, a console window should open. From there, enter "ipconfig /all" and press ENTER. From there, you should be given a list of text. Scroll up to Wireless LAN, and look at IPv4 address. On the right should be a couple of numbers, (xxx.xxx.xxx.xxx) copy this down by right clicking the window, and pressing mark; Highlight the area and hit enter. Don't copy any parenthesis or letters.
Mac
Locate your way to your desktop. Pull up the apple menu under the logo and scroll down to System Preferences; then select "Network" your IP should be on the lower right as "IP address (xxx.xxx.xxx.xxx). Once you have your IP, copy it down.
Linux
Either you use the network diagnose center (depending on distribution), or the terminal with ifconfig. The output should return all your interfaces. Search for inet addr:xxx.xxx.xxx.xxx, copy the xxx.xxx.xxx.xxx numbers down.
Once you have this IP, enter it in the "Output IP / Server IP" or what ever way it asks for where the service points to.
Once you have completed it, find where it says to save/continue/apply. And you have successfully port forwarded. When you run your minecraft server, you have to leave the Server IP field empty in the server properties.
For people to connect to your server, they must use your external IP, which you can find at websites such as IP Chicken. If you don't want to use such IP's, use DynDNS services such as NoIP DynDNS.
Edit: O_O Huge text block.