A "ping" test involves sending a packet of information to a target in order to test whether that target responds and is "alive". Some hardware firewalls, or servers, will not respond to a "ping" test so a failed "ping" test is not a reliable indicator that a server or service isn't working.
Even if a server does respond to a ping test, there's no guarantee the server is working correctly. The ping test suggests that it is switched on, that it's network connections are working, and that the server is responding to pings. It does not guarantee the server is serving web sites or doing what it is supposed to be doing.
Another thing a ping test is useful for is testing out what DNS settings your computer is using.
Sometimes, when you add a new domain to a system, or create a new service for a domain, it doesn't appear to work correctly. Assuming there is no actual problems with a service, the error can be caused by your computer (or ISP) having out-of-date DNS information stored about that service or domain.
A quick "ping" test will tell you what IP address is being used, and if you know what the correct IP address should be you will know whether your computer is connecting to the wrong server or not.