As a child, Dr. Zoidberg aparently went through multiple stages that look like other sea animals, like the sea sponge or angler fish. He went through several stages before growing into his current form. As a child, he was pushed to be a doctor, but wanted to be a comedian. His parents rejected his goal, and continued pushing him to study in the medical field. Later as an adult, he tells his mother he is ready for medical school, and she yells at him for giving up his dreams of being a comedian.
Zoidberg is the nephew to the famous comedian, Harold Zoid, who appeared in black and white hologram sketches much like those of old movies. Zoidberg has many traits of stereotypical Jews, but it is unknown if he even follows any religion. He has three parents on the planet Decapod 10, named Norm, Sadie, and Sam. Two appear to be his real parents, and the other raised him. He has a cousin Zoidfarb, a blue version of himself in an opposite universe (who became good friends with Zoidberg), and several brothers who budded off of him in the earliest stage of childhood.
Dr. Zoidberg works as the doctor for Planet Express, but knows very little about human healthcare. He is often seen writing down notes about human anatomy or characterists. Usually when opperating on patients, they either die, or wind up worse than they were before. Once when he cut Fry's arm off, he reattached it on the wrong side. Zoidberg says that he lost his medical degree in a volcano.
Even though Dr. Zoidberg is a doctor, he is extremely poor. He can only afford one meal a week, and often eats disgusting or uneatable foods. He gets very excited at the mentioning of food, and will eat anything offered to him, and usually he eats garbage.
Dr. Zoidberg is often mistreated in the office. He is shooed with brooms, and getting his pay cut for no reason at all. He is often scapegoated by the other members of the crew for things that go wrong, and comes out of his pay. (Once the ship crashed through the roof of the Planet Express office, and Zoidberg was faulted.)
Dr. Zoidberg has a strange mating practice. His people often start producing intense testosterone, and go into a 'roid rage' state, in which their top dorsal fins stick up. Zoidberg had to be contained, and have his claws rubber banded shut in this stage. He tries to attract a female on the beach, where he wears strange sea plants, and calls weird noises. Being very stinky is also an important factor. Apparently, one the mating season ended, so did his hormone rush (fortunately for him, since the mating process is fatal for his species).
Zoidberg has a horrible stench, which has proved both advantageous and disadvantageous in different episodes. In the episode "Viva Mars Vegas," he was turned invisible, and consequently, everything he ate turned invisible too. Using this power, he played a major roll in a casino heist, by eating the money in the vault and smuggling it out inside himself. But because of his stench, the crew had to use rotten shrimp to cover his smell so that he wasn't detected despite his invisibility.
Throughout the series, Zoidberg has been seen courting a few women. In the episode "A Flight To Remember" he paired up with a jellyfish alien. In the episode "Why Must I Be A Crustacean In Love?" he went back to his home planet for mating season, and tried to attract Edna, a girl he went to high school with. She went out with him, but after learning that all the charming things he had said to her had really come from Fry, she fell in love with him instead. By the end of the episode, she ended up mating with someone else, and died.
In the episode "Stench and Stenchability," Zoidberg met a flower saleswoman named Maryanne, who was not repelled by him, because she was born without a sense of smell. They fell in love, but when she confessed that she would do anything to know what smell is, he offered to give her a nose transplant, knowing that once she was able to smell him, she would leave. But since Maryanne had never smelled anything before, she didn't think Zoidberg smelled bad. She became a garbage truck driver and she and Zoidberg continued their relationship.
"The gymnasium? Excellent, excellent. For some reason I'm frisky as a squid on Tuesday."
"Nobody will ever want to mate with me, not with a puny claw like this! Did you see those other guys? They looked like giant claws with bodies attached!"
"Look at me! I'm Dr. Zoidberg: Homeowner!"
Bender: "You're looking less nuts, Crabby."
Zoidberg: "I'm feeling less nuts, thank you. Because tomorrow I will be depositing my jelly in the cloacal vents of a female...if you catch my drift."
"Oh, the guilt. The UNBEARABLE GUILT!!!"