For the shechitah (ritual slaughter) to be halachically acceptable, both the windpipe and the food pipe of the animal have to be cut. The cut has to be through a certain section of each pipe. If it is made above or below that section, the shechitah is considered invalid. That is called hagramah, which means tilting. The knife was tilted to a point where it became problematic and cut the wrong section.
The windpipe is encircled by rings of cartilage, separated from one another by thin pieces of skin. The highest ring—the one on the side nearest the head—is the limit for shechitah. A cut made above this ring is a problem, but a cut through the ring is okay. In the Mishnah, we're going to have a machloket (dispute) about a cut made through the highest ring, which is okay, but then partially above it. When he put the knife through the highest ring, he put it through at an angle, so the cut finished above the ring.
The Mishnah says that if somebody slaughters an animal by cutting within the highest ring, that is okay. He leaves a hairbreadth of the entire ring attached to the animal's head, and the shechitah is valid. He cut all the way through the ring, very close to the top, but left a portion of the ring attached to the head. That's just like a hair, like a hair deep. So the entire cut is made within this ring, even though it was very close to the top. We say the shechitah is valid, but if any part of the cut is above the ring, the shechitah is problematic. According to this view, the entire cut has to be through the section designated for shechitah. No cut whatsoever. No cut whatsoever above.
The next tanna (Mishnaic sage) is going to be a little more lenient. Rabbi Yosei ben Rabbi Yehuda omer (says) that the minimum he has to leave over is a hairbreadth of most of the ring. In other words, if he cuts most of the way through the highest ring and then finishes the cut above the ring, the shechitah is valid. According to this tanna, he would say that it's enough to make most of the cut in the designated section, but if some of it is above, it's still considered to be...
According to Rabbi Yosei ben Rabbi Yehuda's opinion, shechitah has to be done only through most of each pipe. If you didn't complete the cutting, it's still considered okay. If anything goes wrong after the majority of the pipes are cut, we ignore it because all you needed was the majority.
But according to the tanna kama (first tanna), he also agrees that the majority of each pipe is enough. But he does say that if you continue slaughtering after most of the pipe has already been cut, it's still considered part of that same cutting. Therefore, if something goes wrong after that, even in the last part of the shechitah, when everything was done accordingly, and if you stopped at that point in time, you would be okay. Since it's still part of the same cutting, it still could invalidate the entire shechitah.