The buns are called "Taiyaki", literally means "burned bream". Bream is regarded as a symbol of happiness or celebration, so it would seem that bream was used in this sweet's name.

Dough (mixed with water, flour, egg and a little sugar) is put on the fish-shaped iron plate, then sweet bean paste, custard cream or other sweet stuff are added. You can see the process making Taiyaki at most of shops.
Sometimes Japanese enjoy having the small "discussion" that from which part of Taiyaki you'll start to eat, head or tail. Taiyaki mixes with Japanese daily life well and plays an important role as communication tool among young and old.