One of the two pipes in the neck of a bird, either the windpipe or the food pipe, or both pipes in the neck of an animal, is a situation where his shechita (ritual slaughter) is valid. Even though the shechita of an animal requires cutting both pipes, when we're talking about the shechita of a bird, as long as one of the pipes is cut, it's considered to be a kosher (fit for consumption) shechita. Chachamim (the Sages) explained this as Bedi'avad (after the fact). A person should, of course, try to cut both pipes of the bird. But if you only cut one of them, his shechita is still kosher. In both situations, cutting the majority of the pipe is like cutting all the way through it. You don't need to cut through the entire pipe. As long as you get the majority of it, the shechita is valid.
The next part of the Mishnah will talk about the dam, the blood of an animal that is prohibited to be consumed. Rabbi Yehuda omer, says Rabbi Yehuda, even if the bird was slaughtered properly and the shechita was done kosher, when talking about a bird, you're not allowed to eat it unless the veridin, the blood vessels in the neck, the carotid arteries, are fully, fully drained of blood. The animal is not kosher. Even though, when we're dealing with regular animals, roasting or salting the animal would rid it of its blood, when talking about a bird, a bird is normally cooked or roasted whole. So roasting or salting it wouldn't remove all of the blood. Therefore, we have to cut the veridin to drain the blood. It has to be done—or should be done—while the shechita is happening, in order to make sure that the blood doesn't congeal. There's no need to cut the veridin in the same manner as shechita. You could puncture it. You could just remove the blood. It's not part of the shechita process. It happens at the same time, but you could remove the blood in any way.
Chatzi echad ba'of ve'echad va'chatzi ba'behemah shechitato pesulah. Somebody cuts halfway through one pipe in the neck of a bird, or cuts entirely through one pipe and then halfway through the second in the neck of an animal. The shechita is pesulah because you're cutting only halfway through the pipe. You're not getting the majority of the pipe, so it doesn't count for anything. You have to get the 51% of the pipe, not 51%. Rov echad ba'of ve'rov shenayim ba'behemah shechitato kesherah. Somebody cuts through most of the way through one pipe in the neck of a bird, and most of the way through two pipes in the neck of an animal: shechitato kesherah. The shechita is considered to be valid.
We're repeating this point over here because we're talking about korbanot (offerings), even though we said in the earlier part of the Mishnah that the majority of the pipes have to be cut, each one individually. A person may think that the point of a korban is to really throw the blood. A person would think, since that's the whole goal, I may have to cut the entire pipe in order to make it easier for the blood to come out. But we say no. Even in that case of a korban, we're not cutting most of the pipe; there's enough as well.