What are the important symbols in this book and what do they mean?
I'm reading this book in college and and I'm confused about all these random symbols...my professor says stuff like when Celia was impaled on the fence that the fence represents a penis...it almost feels like my professor is making this stuff up.
As you probably know, symbolism is when an object is used in a story to represent a deeper meaning. It is a common reaction when looking for symbolic meaning in objects, to think that one could choose almost any object in the story and contrive meaning from it. There is, however, a method to what may seem like symbolic madness.
The first step in looking for symbols in a story is to consider the major themes in the story itself, and the author's tone (or attitude) in presenting the themes. Multi-faceted writers of renown literature (of which Eugenides is included) very often embed their stories with multiple dimensions of detail to support their themes and overall message.
As the collective narrators seek to piece together their memories of the tragedy of the Lisbon sisters, one important message to note is the conflict between budding sexuality and curiosity (both for the boys and the girls) and a strict and sheltered environment driven by conservative religious parents. Celia's death on the fence post is certainly intentionally phallic. It is meant to appear as both an ironic demise for this young virgin (who ceremoniously bathed and dressed in white before killing herself) but also insight into the minds of the narrators, who are also fixated on sex. As you continue to read, be aware of other religious details, and consider the sexual message that may behind them.
In addition to sexual and religious symbolism, another prominent symbol you should be aware of, is the idea of boundaries. As objects, these will be presented in the form of actual physical boundaries (curfew, strict dating rules, supervision), however, look for ways of interpreting them on an emotional level as well. First, there is the physical boundary that the boys cannot see into nor enter the Lisbon house. With such limited vision comes a limit on reliability. Consider that the majority of the story is clouded with the boys' imagination and lust. As you note parental imposed boundaries see how these add to the girls' feelings of being pent up (literally, emotionally, and sexually) and lead to rebellion.
Finally, nature and environment also represent predominant symbols in the story. The homogeneous neighborhood (similarity of the houses, yards, and trees) presents a world in which everything is meant to be similar and therefore safe. But as things start to unravel in the Lisbon household, the physical environment begins to change outside as well. The unraked leaves blowing into neighbors yards is just the beginning of the proof that the Lisbons are out of place, physically and otherwise. Natural decay, disease, and pollution creep up around the Lisbon household (note the use of smell imagery) which seems to parallel the sense of decay and pollution happening within the house when the girls are basically locked inside as punishment for Lux's behavior.
Symbolism is not as arbitrary as it may seem. If you keep in mind the author's overall tone and intended message, you will become more aware the depth of meaning in the details.