There's NY State and NY City. Brooklyn is 1/5th of New York City... along with Manhattan, Bronx, Staten Island and Queens. We call them boroughs, but they're really counties. The confusion -- although I don't know why someone told you what they did -- is because the county of Manhattan is named "New York" and Brooklyn is named "Kings" and Staten Island "Richmond" country. These names go back to the English rule of law & geographies. (Where are you from, anyway?)
On its own, you can fit Boston, Cincinnati, St. Louis and Philly INSIDE OF BROOKLYN.. to give you an idea of its land-area.
Areas can be anywhere from 5-min to 45 min. away by subway.. Brooklyn Heights is right beside the Brooklyn Bridge. If you can, look into the areas of Park Slope, too, and Carroll Gardens, Boerum Hill, Cobble Hill, Williamsburg, DUMBO, Prospect Heights, etc. -- again, see a map. (MTV has filmed their Real World in Red Hook, but that's too removed & industrial.) Taxis are easy, coming back to Brooklyn more than looking for one, heading in. And it's cold in NYC now. TIMES SQUARE is in the center of Manhattan. The Statue is on an island, which you need to get to via a ferry that leaves from the very bottom of Manhattan, which is shaped like a canoe, really.