Run the server and don't tell anyone (except whoever you want to play with) about it. Or if you just like the extra protection get hamachi.
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Check the Minecraft Wiki. It has information there.
To make a private server, you will need to go to your server.properties file and change it to whitelist true.
Why would you make a minecraft server to play on it urself? It wastes RAM and time.
It does not cost anything to get a minecraft server. You simply download the Minecraft_server.exe from the minecraft website download page. Then, you run it to make your own server.
In server.properties Edit The "motd= Server" And Make It Whatever You Want