Long Island is a long, finger-shaped island right off the shore of New York City to the southeast. It is east of Manhattan. It is surrounded by the Atlantic Ocean. If you look on a map, you will see it jutting out to the east. There are a lot of Indian named towns on Long Island (Patchogue, etc.). There are four counties on Long Island. It is kind of a bedroom community to New York City. Many people that live there work in "the city."
Tom and Myrtle's love nest was in Manhattan, which is one of the five bouroughs of New York City. If you look on a map of New York state, you will see Manhattan on it. Manhattan is where the Twin Towers used to be (lower Manhattan). East and West Egg were both located on Long Island.