Roppongi has Great Nightlife
When it comes to late night clubbing, Roppongi is high up there with a wide variety of classy restaurants, late night clubbing, and fancy hotels. It also tends to attract a lot of expats/tourists, so if you're looking to go out to a place where you can speak English, Roppongi is a great place to check out.
I highly recommend Bar Mania, it's on Roppongi Dori, affordable drinks, music is kept to reasonable levels, and the staff are very friendly. Be sure to ask for Walter-san, he makes great drinks.
I also recommend checking out Tokyo midtown, it's a lovely park in the center of Roppongi that often has special events during the day.