For the most part, the Planet Express crew seems to actively dislike Zoidberg, and Professor Farnsworth and Hermes aren't exactly warm fuzzy balls of sympathy and compassion. They have to be employing him for a specific purpose - Amy is there for her blood type, the three main crew have their purposes given in the first episode, Hermes to run the company, Scruffy to clean (presumably) - but what is Dr. Zoidberg's actual job at Planet Express?
The Wikipedia page appears to be out of date.
The latest season reveals that Zoidberg is in fact employed for a very specific purpose:
To kill Dr. Farnsworth should he develop symptoms of hypermalaria, due to the gruesome death it entails otherwise. While this proved ultimately unnecessary, due to the Professor not being infected with hypermalaria at all, but rather with Yetiism, he is likely kept on even after this, out of gratitude.
Besides all of this, he is a moderately competent knower of non-human biology, which is occasionally useful.
The very first episode gives ample evidence that he is employed by Planet Express.
We know that he is very poor, and expects doctors to be poor (From the episode where Leela begins dating a doctor). Thus, he's not very well paid.
I suspect, but have no canon confirmation, that he is employed by Planet Express because he will give the results the management wants from any given procedure (his extreme pliability is demonstrated in How Hermes Requisitioned His Groove Back).
It is likely FAR easier for this interstellar shipping company, which most likely must keep all medical records up to date and provide documentation of things like immunizations, biohazard clearance, etc to have a doctor on staff rather than bribe a different one each time they need falsified paperwork.
Season 6, episode 18, "The Tip of the Zoidberg" explains how the Prof and Zoidberg met and why Prof Farnsworth won't fire him.
As explained in the episode titled "The Tip of the Zoidberg," he has a pact with the Professor to kill him once the effects of hyper-malaria set in.
I thought it was just a spoof on Star Trek, where a medical officer figures prominently in the cast of characters.
Zoidberg and the Professor are old "war" buddies, and the Professor thinks himself to have caught Tritonian Hyper Malaria. Because of this, he wants Zoidberg to kill him when the symptoms begin (as shown in "The Tip of the Zoidberg). However, the Professor really just has Yetism, so Zoidberg is really kept around from then on because he saved the Professor from that and cured him.
zoidberg did say in an episode something along the lines of printing a medical degree.....and in the episode where leela suffers "squidification" zoidberg says "lets get you to a real doctor"......though when they're hunting the tritonian yeti mom says "you want an alien hacked to pieces, he's the best we,ve got....in the budget category" which makes me think the med school was probably a not good one and in the same episode Conrad suggested that if zoidy was to inject the toxin from the yeti that it would make him a double yeti......guess what Conrad was right.
Zoidberg is the cheapest (free) doctor available and Planet Express is required to have one doctor on staff.