What name do I use while filing for a trust Trust name or trustees name?

On which form and where? On Form 1041 and associated K-1s, you would enter both.

If you are entering information from a K-1 you received, you would enter the name of the trust and its EIN.

On the "Personal info summary" it's asking me personal information, but I'm filing for a trust.  

First name, last name, my date of birth, marital status, etc...

Do I put my personal name in the "first and last name" section?  I'm filing on a Business Turbo Tax application.

Also, it requesting a social security number instead of a EIN#. Do I just put the EIN in the SS section?