Considering the enormous amount of college students in the area, there isn't a specific area where you will find a concentration of college students. The city of Rochester is 5 or 10 minutes away from campus, and 20 by bus, so having a car comes in handy. Eastman School of Music students have the city at their fingertips, and there are some good coffee shops and restaurants there, but for the most part the city is fairly quiet; everything tends to close early. East Avenue and the Gibbs Street area have the coffee shops and theaters, and down Alexander Street are some nicer restaurants and bars. Down in the St. Paul area are some clubs as well.
There are about seven suburbs of Rochester that surround the campus and the city. Henrietta in particular has tons of stores, malls, and restaurants as well, but they tend to be more of the franchised businesses, which is convenient if you need big-box outlets like Wal-Mart, Home Depot, or mall stores. A lot of the suburbs outside the campus are just quiet residential. If you go a few hours in any direction, however, you can find some fun. Niagara Falls is only an hour and half away, and Toronto is a few hours away. Students tend to like going up to the Falls for a night-it's a fun day or night trip. Being in upstate New York with lots of colleges around also has its advantages. If there are concerts or people you know, you can hop down and visit easily as well.