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: 
You can't. You have to send an email and wait for an answer to verify that this address exists.
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.