In the first episode of Fear the Walking Dead, Madison clearly gets bitten by a zombie. Why did this bite not kill her like the other victims? Is she maybe a type of a cure in the apocalypse?
Your question is flawed for a number of reasons - she wasn't bitten, but she is infected (like everyone else), and the creator of the franchise is strongly opposed to the idea of any cure being found.
Was Madison Bitten?
Absolutely not. She was attacked by Calvin, but Travis and Nick saved her.
Nick, Travis, and Madison get in the pickup truck and back up out of the aqueduct. Travis sees Calvin walking towards them. Madison and Travis get out and approach him, despite Nick's intense warning not to, and see the massive amount of blood on his short. He tries to bite Travis but Travis fights him off long enough for Nick to back the pick up truck over him. Despite the massive impact of the truck, Calvin rises up again. He's a zombie. Nick floors it in drive and slams into zombified Calvin and launches him down into the aqueduct. Travis, Madison, and Nick look on and see that Cal's body is still moving. "What the hell is happening?" Madison asks. "I have no idea," Travis says as he turns to Nick.
Assuming Nick really has been hallucinating this whole time, they leave the scene only to encounter a shambling body in their exit tunnel. They stop the car, recognizing it may actually be Calvin. Madison and Travis step out of the car with Nick yelling at them to stay back—he knows what’s happened to his onetime friend, as it happened to Gloria, too.
He’s proven right as Cal tries to attack and bite Madison, but Nick comes to the rescue by ramming the undead Cal with the car not once, but twice, as he won’t seem to stay down. But finally, they seem to put an end to whatever the thing was that attacked them, and they ask that fateful, if not obvious question.
Does that mean she isn't infected?
Not at all. She definitely is infected - everyone is. This is true of both series, as well as the comic books and print novels. I was curious as to whether the "everyone is already infected" rule was in play in the first season of Fear the Walking Dead, and I got the answer in the finale: people who died in the military hospital without having been bitten were turning into zombies, so the medical personnel were forced to use a captive bolt gun to destroy their brains as soon as the patients died. Thus, everyone was infected by no later than 9 days after the events of the season premiere.
"They all come back. They all come back."
- Dr. Exner, Fear the Walking Dead, season 1, episode 5, explaining why Grizelda's brain must be destroyed despite the fact that she died without being bitten.
“I've seen what it does. The bites don't turn you, but the infection is not treatable. The infection kills you like anything else. Then it happens. It doesn’t matter how you die. You come back. We all come back.”
- Liza, Fear the Walking Dead, season 1, episode 6
The franchise's creator has said that bites don't infect you with the zombie pathogen, they just kill you. You're already infected.
The rule is: WHATEVER it is that causes the zombies, is something everyone already has. If you stub your toe, get an infection and die, you turn into a zombie, UNLESS your brain is damaged. If someone shoots you in the head and you die, you’re dead. A zombie bite kills you because of infection, or blood loss, not because of the zombie “virus.”
- The Walking Dead Wikia, quoting Robert Kirkman
Immunity or a Cure?
No, she isn't immune (no one is immune), and she doesn't have the cure. The creator of The Walking Dead and Fear the Walking Dead, Robert Kirkman, has dropped some pretty strong hints that there will never be a cure.
If we're going to be witnessing the beginnings of the zombie plague, does that mean Fear the Walking Dead will address its actual origins? No way, as Kirkman explained. "I still maintain that’s not an important aspect of the story," he said. "I think watching Fear the Walking Dead, you’ll get a bigger picture of the world, but as far as digging down to actually find the smoking gun and realizing what that causes, it’s really just unimportant to the overall story. You’ll see when you start watching the show. If we were to do a spinoff of The Walking Dead and it was about a bunch of scientists that were working to find the cure and finding out the origins, that would bore me to tears."
The crowd offered loud boos when a fan asked Kirkman if the survivors would find a cure for the zombie plague. Kirkman left no doubt about the producer's intentions.
"If we were into finding the cure, this would be a completely different show," Kirkman said. "This is about the struggle to survive. That's the story we want to tell. It's much more engaging."
I know I am extremely late to the discussion. Yet I'd like to point out Madison was bitten however the bite was over at least two layers of clothing. It more than likely would have been very superficial, caused some bruising and most certainly some pain - hence her exclamation, "ow!"
Yes, everyone is infected therefore she to is infected. However, her demise will not be just yet. With any luck there will be a mash up down the line with Fear the Dead and The Walking Dead.
Look, I have the DVD, she was clearly bitten, and shows the bite wound.
What's more, the next 24 hours, they seem to pay close attention to her, and the promises she extracts thinking her time is running out. Perhaps they attempted to rewrite it, or just drop it, and keep her in.
If however, it helps anybody else with continuity issues (as it's bugged the hell out of me) I think I found a solution. Calvin (as of when he bit her) had JUST turned himself. He likely had a nasty, human mouth. But NOT the mouth of a being dead for weeks, or months and not nearly as infectious. She did seems ill, and seemed to take meds herself, and perhaps pulled through.
Sadly, she should have opened her mouth and told Liza she had a chance; but given her agreement with Daniel's ruthlessness, maybe she didn't want to compete?