May I ask your name?

Literal

Name [topic-は] what [quotative-と] honorable-say [polite-ますか].

おっしゃる is the sonkeigo (honorific) form of 言う — used when a respected person speaks. 何とおっしゃいますか literally 'how do you (honorably) say?' = 'what is your name?' Very polite, formal introduction.