Question: How can i make a script like verify-email.org ?
I told in the school, that this is not a problem. But now, I searched the web, and found nothing?

E.g. test123@hotmail.com

* I don't want only check if the domain exist of the E-Mail address. I want check, the whole E-Mail, if it exists on Hotmail.com?

Thanks.

Answer: 
The only clear way to check an email address is valid is to send it an email. If it isn't valid you will get an undelivered response.
Like my predecessors wrote, you can only completely veryfiy by sending a mail to the address.

However, you could use DNS to check if the domain exists and a MX (Mail Exchange) record is associated with it. 
It's not a complete solution, but maybe a step that brings you a little further.