Norman is a really nice father, like Bea's mother. He also has a sense of humor. He is a cheerful fish as well.
Mr. Goldfishberg first appeared when he gave Bea's mom her favorite dessert, Fish Flake Pie. They ate it how "adults" eat it grossing out Bea ("Bea Becomes an Adult Fish"). He then appeared when he and Mrs. Goldfishberg were chaperones for the student council dance. Bea created a fake school dance because she thought they will embarrass her, causing her to lose her crown. He thought the fake dance was real along with his wife and later he wanted to dance with Bea. They did, and when Bea got called back to the real dance, Oscar had to dance with him. They later figured out the dance was fake and Bea told them that even though the dance was fake she did not have fake fun with them. They told her they did not have fake fun also, and Bea crowned her parents instead of herself ("Queen Bea").
His dance moves got Mrs. Goldfishberg to marry him ("Queen Bea").
Norman's name could be a reference to Not Norman: A Goldfish Story, a book that his voice actor and creator of the series, Noah Z. Jones, illustrated.
It is possible that the creators of Fish Hooks made him a teal-blue color to show that he is old. In real life, goldfish turn more of the color that Bea is, and teal-blue is their juvenile coloration. In other words, Bea and her parents would have swapped colors in real life.
In "Bea's Commercial", it is shown that he wants Bea to be into trivia instead of being on a commercial.