First off, do not put your computer on a public IP. Ever. For port forwarding, you want it to be inside your network with something like 192.168.0.10 (or whatever your internal network addressing scheme is). You'll want to make a port forwarding rule in your router that anything your router receives on port 25565 gets forwarded to this internal IP. If you give the make/model of the router, we might be able to find some instructions specifically for that router. I also wouldn't recommend turning off your OS firewall...just create an exception for it.
You can internally test to see if your minecraft server is accessible from another internal computer by typing "telnet 192.168.0.10 25565" from a DOS command prompt (using the IP address example above...substitute for whatever you set it to). If the screen goes blank, then your server is running properly on the internal network (it's answering that port connection request).
If you put your server straight on the Internet, it allows people to connect to it and run scanners to find vulnerabilities which they can use to do whatever they want with it (DDOS attacks, mail relay for spam, using your server as a warez FTP server, a relay for any all sorts of illegal activity, etc). I've seen hackers find vulnerable servers that were set up during the day and compromised by the next morning.