His complete name is Edward Anthony Masen Cullen. The Cullen last name is adopted from Carlisle because he's his adoptive father after Edward's parents died.
Carlisle knew him when he was dying of the Spanish Influenza.
Before Edward was transformed into a vampire his full name was Edward Anthony Masen. But when Carlisle realised that Edward was going to die from Spanish Influenza, he changed him into a vampire and Edward adopted Cullen onto the end of his name.
And the actor's full name is Robert Thomas Pattinson.