Roppongi is home to the famous "Roppongi Hills" area. Its got a huge active club/bar scene and is crowded with young foreigners though of course the majority of visitors are Japanese. I'd say in the 3 visits I've made to Japan, most of my night outings were in this area which surprisingly is bustling on any night of the week. Option 2 I'd say is Shibuya which has much fewer foreigners and a larger hip-hop scene though whatever you fancy I'm sure you can find it in Shibuya. Shibuya seems to me to be where are the young fashionable Japanese flock to. If you walk around for an afternoon you'll see enough "eye-candy" to make you wonder why you live in the US. The night scene is just as fun.
Roppongi's main night club/bar road is the road that leads straight to Tokyo tower. If you like techno or just want to stop by a bar on your way to the main street, there is a bar on the way called "Heartland." Its a popular standing bar that serves a pretty wide selection of drinks including Absinthe (@145 proof) as well as a self-brewed beer that's actually purdy good. They play exclusively techno though so you've been warned/informed.
Well this night we went out to 911. 911 is a "Cigar and Champagne bar" by name. On "Ladies Night" (I think that's Thursday or Wednesday) they serve free champagne to the ladies. I personally find some fun in sitting back listening to music ordering a martini and trying one of their cigars [note I'm not advocating you to smoke a cigar or drink alcohol esp if you are underage. Furthermore, if you have never smoked a cigar, unless someone explains it to you or shows you how, you shouldn't be trying. Its not something you inhale and you can seriously damage your body and lungs with the carcinogenic smoke. That being said, you are still in risk of mouth cancer. Stepping of my box...].
The bar has probably the biggest selection of drinks I've ever seen. They have like 10 different types of Martinis, and for any base probably just as many types. Whatever you fancy they have. The music is an excellent mix of old and new music, hip-hop, latin, techno, whatever you can name. They do a good job of mixing it up and keeping the environment happening. Its get paaaacked on weekends though. There's no cover so you should plan on paying 1000Yen for a seat if you go with friends and don't want to stand the whole time. The "VIP" room can sit about 8 comfortably and cost 10,000 yen so its an easy split if you don't mind paying a few extra yen.
One last note... there's a famous chain of clubs called "Gas Panic." They play great music but its so crowded with foreigners you can't move, you can't talk, and you get stepped on a lot. But if you've never done it, its easy to find. There are two in Roppongi. Ask anyone and they can direct you on your way. That being said, I don't recommend it.