This is one of the oldest streets in New York City, if not the oldest, since its origins go back even before the days of New Amsterdam. It started out as the Wickquasgeck trail, used by the Indigenous Peoples to cross Manhattan island. It was turned into a street by the Dutch, who called it "BreedeWeg" - literally 'Broad Way'.
In modern day New York, the street begins at the Bowling Green and extends across the entire island of Manhattan, through the Financial District and past Times Square and Columbia University.
Of course, at the time you're visiting, Broadway ends at the city common. The most famous stretch of the street - the Theatre District - is still wilderness, so you're more likely to catch a bullet there than a show.
Which in some ways is preferable.