Gyoza is a dumpling stuffed with vegetable, seafood or meat. It is originally from China but Japanese adopted the Chinese dumpling to their own style after WWII.
Gyoza can be pan fried, deep fried, steamed or boiled but the most common and popular one in Japan is pan fried gyoza which is called Yaki-Gyoza in Japanese. On the other hand, Chinese dumpling is mostly boiled one and Chinese fry the dumpling after boiling it but Japanese boil right away from the raw one.
It should be eaten with soy sauce, rice vinegar and Rayu(Japanese chili oil). You mix the soy sauce, vinegar and the chili oil on the small plate and dip the gyoza on it. Chinese do not eat their dumplings with white rice but Japanese love to eat gyoza with white rice.
There are two major Japanese cities which known as a city of gyoza. The first one is Utsunomiya, which is the capital of Tochigi prefecture, north of Tokyo, quite close to Nikko(From Tokyo, you can take Tohoku shinkansen to Utsunomiya and change there to Nikko line). The other one is Hamamatsu, the largest city in Shizuoka prefecture, which is between Tokyo and Nagoya on the Tokaido shinkansen. These two cities' rivalry are quite heated up as Gyoza war.
If you do not have any chance to visit these cities, just enter a Chinese diner or ramen noodle bar which normally offers gyoza as a side dish. If you really want to try a nice but cheap one, please visit restaurants as like Gyoza no Manshu or Gyoza no Ohsho, which have franchise restaurants throughout Tokyo area. By the way, the price is roughly EUR 2 for 6 dumplings!