In the tenth chapter of Tractate Zevachim, we continue to discuss the hierarchy of the offerings - which offering takes precedence over another and which is offered first. Mishnah 4 begins: "Ha'ofos kodmin lamenachos, mipnei shehen minei damim" - bird offerings take precedence over meal offerings, because they are blood offerings - bird offerings receive priority over meal offerings, because they contain blood that is offered, and blood is the essential mechanism through which atonement is achieved. Even though there are meal offerings that come to atone for a sin, as will be explained shortly, the primary path to atonement is through blood, and therefore blood offerings are more powerful and important, and they take precedence.
"Minchas chotei kodemes leminchas nedavah" - a sinner's meal offering takes precedence over a voluntary meal offering:
When a kohen has two meal offerings before him, and one of them is a sinner's meal offering that comes to atone for a sin, it takes precedence. This refers to someone who became obligated in a sliding-scale offering (korban oleh veyored): a person who transgressed one of the sins detailed there - such as a tamei person who entered the Beis Hamikdash and the like - and cannot afford to bring an animal or even a bird, brings a meal offering. This meal offering takes precedence over a meal offering brought voluntarily, which a person volunteers without any special reason, because it comes to atone for a sin.
It is worth noting that a sinner's meal offering is nothing more than fine flour and water: no oil is mixed into it and no frankincense is placed upon it. Seemingly, one might have thought that the voluntary meal offering is more important and holier, since it contains the ingredients of oil and frankincense; but this is not the case, for the atonement for the sin is infinitely more relevant.
The meal offering of a sotah:
There is another type of sinner's meal offering - the meal offering brought for a sotah, and here the matter is more complex. A sotah's meal offering does not come to atone for a sin, but rather it is part of the process to clarify if there is any sin here at all, as perhaps she is completely innocent. On the one hand, she is associated with sin, for she certainly acted improperly to reach the situation that a sotah is in; on the other hand, she is only a suspected sotah and not a definite sotah, and her meal offering does not necessarily come to atone for a sin.
The question is, therefore:
If the reason for the priority of a sinner's meal offering is that it comes against the background of a sin that occurred - then the law would also apply to the meal offering of a sotah.
And if its priority stems from the fact that it comes to actually atone for a sin - this would not be relevant to the meal offering of a sotah.
The Mishnah leaves the question unresolved and does not decide the matter. Therefore, the Rambam rules that if a kohen has two meal offerings before him - a sotah's meal offering and a voluntary meal offering - he may offer whichever one he wants, since there is no decision as to which of them takes precedence.
"Chatas ha'of kodemes le'olas ha'of" - a bird chatas takes precedence over a bird olah:
Bird offerings come in most cases as a nest (ken), as a pair of birds, where one of them is an olah and one of them is a chatas, and the rule is that the chatas always takes precedence over the olah. This is also true at the time of consecration: the nest can be consecrated as one, both birds together, and the first one to be taken and identified for its final designation - must be identified as a chatas.
This rule actually applies beyond bird offerings, and even to other offerings in general: an animal chatas takes precedence over an animal olah - whether a cow, a goat, or a sheep. The source for this is that the verse explicitly states regarding a bird chatas: "Vehikriv es asher lachatas rishonah" - he shall offer that which is for the chatas first - meaning the bird that is a chatas takes precedence. However, the verse continues and says: "Ve'es hasheni ya'aseh olah" - and he shall make the second an olah. Seemingly this is obvious, that if the first is a chatas, the second must necessarily be an olah. What, then, do these words come to teach? To teach us that the law of the priority of the chatas does not apply only to birds, but also to animals. Therefore, in any scenario where one must choose, the chatas is offered before the olah.
We should compare this halachah to what we learned earlier in the chapter regarding the limbs of an olah and the emurim of a chatas that are ready to be brought up on the altar:
Bringing up on the altar: When a kohen has before him the limbs of an olah and the emurim of a chatas, and the question is what to bring up on the altar first - the olah comes first. However, this law applies specifically to two parts that are ready to be offered on the altar.
Offering the entire korban: When a kohen has a chatas and an olah to offer, the rule is that he processes the chatas in its entirety, which includes offering its emurim, and only afterward does he process the olah in its entirety and bring its limbs.
In summary: In this Mishnah we learned the continuation of the hierarchy of korbanos: bird offerings come before menachos because they involve blood, and blood is the primary means of atonement; a minchas chotei comes before a voluntary minchah even though it lacks oil and levonah, because it comes to atone for a sin; a minchas sotah remains in doubt, and therefore there is no definitive ruling between it and a voluntary minchah; and a bird chatas comes before a bird olah, a rule that is derived from the verse regarding an animal chatas and an animal olah as well.