(Mishnah 1): Chatas ha'Of is offered below (the Chut ha'Sikra, i.e. a divider between the lower and upper halves of the Mizbe'ach. The blood of) Chatas Behemah is offered above;
Olas ha'Of is offered above, and Olas Behemah is offered below;
If any of these was done in the wrong place, it is Pasul.
A Chovah (an obligation to bring birds, e.g. for Taharas Zavah) is one Chatas and one Olah. (From here, all Korbanos we discuss are birds);
Nedarim and Nedavos (vows to bring a Korban) are all Olos.
Question: What is a Neder, and what is a Nedavah?
Answer: "It is Alai (incumbent on me) to bring an Olah" is a Neder. "This is an Olah" is a Nedavah.
Question: What is the difference between Nedarim and Nedavos?
Answer: There is Acharayus for a Neder (if the Korban dies or is stolen or lost, he must bring another), but not for a Nedavah.
BIRDS THAT BECAME MIXED
(Mishnah 2): If a Chatas became mixed with Olos, or vice-versa, even if one became mixed among multitudes, all of them must be left to die. (We do not know where to offer each bird. The Mishnah procedes to discuss Chovos Stumos, i.e. it was not specified which will be the Chatas and which the Olah. 'Three Chovos' refers to six birds, three of which must be offered for Chatas, and three for Olah. The underlying principal behind the coming Mishnayos is that we may not offer a number of Olos (the same applies to Chata'os) greater than the number of Chovos they come from. If more were offered, all the extras are (Safek) Pesulim.)
If a Chatas became mixed with (e.g. five) Chovos, we may offer (like Chatas) from the mixture the number of Chata'os in the Chovos (five). (Even though one more Chatas should be offered, if we select six birds, perhaps all are from the Chovos, and only five may be offered for Chatas. We cannot offer even one Olah, for any bird we pick could be the Chatas that was mixed in.)
For exactly the same reason, if an Olah became mixed with Chovos, we may offer from the mixture the number of Olos in the Chovos.
(Mishnah 3): This applies when Nedavos (Olos -- the same applies to Chata'os) became mixed with Chovos (Stumos). If Chovos became mixed with each other:
If Rachel and Leah each had one, two, or three (or any other number of) Chovos, and they became mixed, only half of them are valid (i.e. may be offered, even without knowing whose they are. A quarter of the total number of birds are offered for Chata'os, and a quarter for Olos. Even if all belong to one woman, it is not more than she was obligated. If we would offer more than a quarter for Olos (or Chata'os), perhaps all were Leah's birds, and this is more than the number of Olos (or Chata'os) that she was obligated, and the excess (over a quarter) had to be Chata'os (or Olos)!)
If one woman gave one Ken, another gave two, another gave three, another gave ten, and another gave 100, the smallest number (of Chovos given by any woman) are valid. (The above reasoning shows that we may offer this many, but not more.)
This applies whether all the Chovos are for one Shem (obligation, like the next Mishnah explains) or two (or more), whether they are for one or two (or more) women. (If a woman bought birds for one or more Chovos, and later bought more birds for other Chovos, each set must be half Olos and half Chata'os, just like birds brought by different women.)
(Mishnah 4): The case of (one woman, and) one Shem is when all are for births (a poor Yoledes brings a Ken Chovah), or all are for Zivos. The case of two Shemos is when she brought some for births and others for Zivos;
The case of two women and one Shem is when Rachel and Leah both gave Kinim for births, or both gave for Zivos;
The case of (two women and) two Shemos is when Rachel gave for births, and Leah for Zivos.
R. Yosi says, if two women puchased their birds together, or gave money together to a Kohen (to buy and offer their Chovos), the Kohen may pick any birds he wants for Chata'os (half the total number), and the others for Olos, whether they are for one or two Shemos.