An egghead is a smart, nerdy person. You might call your studious older brother, who always seems to have a book in his hand, an egghead.
The word egghead is a very informal, slightly insulting name for an intellectual. College libraries are full of eggheads, academics who are researching and writing and thinking about their scholarly pursuits. The word originally meant “bald person,” but it came to mean “smart person” in Chicago-area slang, particularly among newspaper reporters around 1918.