Ok, on that website, they can test your open ports. for example, if you wanted to host a web server inside your network, you would have to configure the internet router to pass traffic on that port from the public interface to the private interface of the web server. When you have the configuration done & before you publish the website, you may want to test to make sure that it works.
You can use their online tool to see if they can make a successful connection to the target port. In the case of web services, the most common port is port 80.
Usually you don't want to expose your ports to the internet, since some bad people may try to take an advantage of your system and break into it. You may loose your personal data stored on your computer, passwords to your online accounts, physical hardware loss, etc.
To protect your system from all these issues, most systems come with firewall settings on. Meaning any services on your machine that are not explicitly allowed to communicate with outside world would be prohibited to do so.
To check if you have any open ports exposed to outside world you can run port scan.
There are numerous sites that can scan your system for free.
Here one of them GRC Internet Security Detection System
If you can find any open ports with scan - your firewall software doing its job :-)
If you have an application that should have port open but you don't see it, you can run a **netstat -a ** program from command line prompt. Netstat will show you all active/established connections, and all open/listening ports on your system.
Use caution when exposing your system ports or you may get more than you bargened for.