There isn't much "night market" out all year around in Japan. Night market in Japan is out seasonally, and it is called "Ennichi".
Learn more about Ennichi:
For nightlife other than that, I think it is not much different from other countries. We go out to bars or Izakaya (Japanese pubs) and have food and drinks with friends, etc. Izakaya may be unique in Japan. They offer not only alcoholic drinks but many places put much effort on food, too. So you can enjoy delicious Japanese dishes and alcoholic drinks (of course, Japanese sake!) together.
Japanese beer, wine, whiskeys are also really good. Recently, craft beer is getting popular and noticed in Japan and there are many craft beer bars/pubs. In such places, you can try various craft beer from all over Japan.