1. A person who is considered intellectually gifted in the field of academics. "Egghead" is usually used as college-speak to describe a brainiac.
2. A person's whose head is shaped like an egg.
Most people however, will use this word interchangeably as a pun. It has also been known that people whose heads are shaped like an egg are usually large at the top, which explains the larger brain-size.
Jackie will probably do well in that PhD program. She has always been known to be a bit of an egghead from elementary school all the way through college.