Let's break it down:
* 아버지 (abeoji): father
* 의 (ui): possessive particle, indicating "of" or "belonging to"
* 성함 (seongham): name (formal, respectful form)
* 은 (eun): topic marker
* 무엇 (mueot): what
* 입니까 (imnikka): polite question marker
So, the phrase is asking about the name of the person's father in a respectful and formal way.