LAN means Local Area Network. This means any device on your local network (or, as you said, "the same internet").
WAN means Wide Area Network. This means any device not on your local network.
You are looking to set up a WAN server, which you can find steps to set up on the Minecraft Website. But to do this you not only have to set up the server, but also you have to port forward, which can be a hassle.
If you don't want to set up a WAN server, you can still use special VPN services to help, such as LogMeIn Hamachi.
I believe that since I included LogMeIn's product in my answer, I have to announce that I am not affiliated in any way with LogMeIn