A domain is not required, all you need to have is a VPS or a dedicated server with apache and mysql installed, and put your php script in the www directory.
FTP will only be used to move your script from your pc to the server, using softwares like FileZilla.
You must open a socket listening to a certain port on your php script, and update the scores on the database when receiving the data.
You can use the TCP sockets from Matrix1 Utils to use sockets on the DBPro side.
If you want to make some practice before to rent a server, you can download XAMPP, this software includes phpmyadmin which is useful to manage your database without having to create and edit everything with sql commands directly.
Remember that the php script on the server must be called to start, you can do that directly from your browser calling like "http://myvps.com/MyScript.php" or calling it with a cronjob.
Warning: Calling multiple times (like every 10 minutes with a cronjob) php scripts with sockets code inside may lead to memory leaks on your server, to avoid this on the cronjob call instead a "check script" that checks if the main script is alive, if not it calls it to start it. When I did my tests, RAM was decreasing by 5MB everytime I was calling the main script.
But this is a step further so it's something you should worry about later.