Japan has some of the best food on the planet. Foreign cuisine is very popular, particularly Korean, Chinese and Italian, and foreign restaurants abound everywhere and are much the same as the restaurants you would get home. Often the menu will be in Japanese only, so you may need to get help from the waitress as to what each dish on the menu is. There are also plenty value for money family restaurants such as the Gusto's and Denny's chains. With these, although the "home cooking" element is certainly NOT there, neverthless they are cheap and you can usually get unlimited tea/coffee for a few hundred yen. Lastly, there are the fast food restaurants such as Macca's and KFC, basically the same as what you get back home, albeit with somewhat smaller portions.
Japanese style restaurants
This is what you come to Japan for, not to eat that crappy fast food. So break out of that comfort zone and try something new! There are many different styles to the ones below, but this will give you an idea.
More like a bar/restaurant hybrid, Izakaya are probably the most popular type of restaurant for Japanese people, and foreigners usually can't keep away once they have discovered them. With an izakaya, all of the dishes are small and are designed to be shared between all the people at the table, so rather than ordering one dish each, you order lots of small, dishes and try as many as you can. You usually have the choice of normal table/chair seating, or you can be brave and eat Japanese style where you sit on the floor around a long, low table.
In Japan, sushi bars are often much cheaper than overseas (often only ¥100 per dish), and the sushi is fresher. You can either sit at a normal table, or at a counter where you order as you go. There are also many "kaiten zushi" restaurants, where the sushi goes around on a conveyer belt and you pick off the ones that you want.
There are 3 major types of noodles in Japan; udon, soba and ramen, and if you get the chance you should try them all. They are usually served as a soup, with various flavours and add ons in the bowl. Many Japanese people have noodles for lunch.
Actually originally a Korean dish which the Japanese have modified and some would some improved, Yakiniku is basically Japanese style BBQ. You sit around a table which has a grill built into it, order your meat (thinly sliced so cooks quickly) and vegetables and cook them yourself. Needless to say Yakiniku goes great with beer. It can work out expensive as you will find yourself ordering more and more as you just cannot get enough of the exquisite taste.
A “hot pot” dish where you are given a large bowl filled with a soupy stew. As in Yakiniku it is placed on top of a gas burner on your table, and you are given your prepared ingredients such as thinly sliced meat, vegetables, mushrooms, tofu and noodles. You then put these yourself into the put and you take them out and share them around as they are cooked.