Best area in Tokyo for night life

What is the best area in Tokyo for the night life?

Depends what you mean by "nightlife".

Shinjuku has many bars
Shibuya has quite a few clubs
Roppongi is more westernised and has clubs

There are a few other areas that some of the locals can elaborate on (Nishi-Azabu, Daikanyama, etc.)

Dance Clubs
http://whereintokyo.com/dbinx/club.html

Jazz Clubs
http://whereintokyo.com/dbinx/jazz.html

Gigs
http://www.tokyogigguide.com/
http://www.clubberia.com/en/

Enjoy your trip!

Roppongi is a hole.
Shibuya is nice, the cool areas are shimokitazawa and koenji.
Tokyo like most cities in japan doesnt work like a normal city though, it tends not to have special night life areas, clubs and stuff are spread out amongst the rest of the city. You have to find a club you want to go to then go there, just wandering the streets in the centre of town and going to a random place as you can do in Europe doesnt work in tokyo.

Shinjuku - hands down , it has nightlife to cover nearly every taste :)

You might find Wikitravel helpful with this kind of query - while it's not as detailed as this site in most regards, there is a section on each area giving examples of nightlife and bars, specifically.

I went to Tokyo this last September, I went to roppongi twice the first time I had fun. Jumping around gaspanic, new luxito and some other place I forgot the name. It kind of has that college town feeling like San Diego /Santa Barbara if your familiar with California.
Majority of people are friendly but there's a couple that don't know there limit and get to drunk.
Shinjuku is great to just go grab a beer and relax, of course with food everywhere. I went there during the week since it always seemed to have a lively atmosphere.
I also went to club womb in shibuya. Probably my favorite place while I was there. Great sound system, lighting And Dj's (if you like house music) I got the sense from womb that people just go there to dance, didn't see many people standing against the wall.
Over all I had a great time partying in Tokyo.

What was the dress code like at the clubs in tokyo? Do you have to wear dress shoes or can you go in with chucks like those type of casual shoes that fits your style/outfit? I'm from LA myself do you know the kind of dress code at the hollywood and LA clubs. No sneakers of course. Jeans. Dress shirt and some casual shoes. I'm asian by the way if that helps.