You can check your public IP on the server with the following commands:
wget http://ipecho.net/plain -O - -q ; echo
or
curl ifconfig.me
To know all tyhe interfaces on the server you can use:
ip addr
From a service like //http://viewdns.info/ or http://www.fifi.org/services/traceroute you can check the IP corresponding to a domain name and verify if it's the same as the public IP on step 1.
A server in a private network can be in public internet opening a port in the router that connects that server to internet. So, if you want to use that server as web server you just open port 80 and redirects everything coming to port 80 to the private IP of the server.