Here we'll talk about three rules about when two korbanot (offerings) are about to be offered, which one goes first. Mishnah says: “Whatever is done more often, then the other one comes first.” In other words, if two offerings are waiting, the one that's brought more often goes first. If you have a korban tamid, which is brought every single day, and a korban musaf, which is only brought on Shabbat, Rosh Chodesh (new month), and on Chagim (festivals), halacha (Jewish law) is that the korban tamid would go first because tadir v'she'eino tadir, tadir kodem, whatever happens more frequently goes before the infrequent ones. That's a rule not only for korbanot, but for all types of mitzvot (commandments).
Another rule: Whatever is holier comes before the other. If two korbanot are waiting to be brought, the one that's holier is brought first.
A third rule, which is about what we've been talking about in this masechet (tractate), is Par HaMashiach u'Par HaEdah (the anointed priest's bull and the community's bull). If you have the par kohen mashiach (bull of the anointed priest) and the par he'elem davar (bull offering for an erroneous ruling), which are both standing ready to be offered, Par HaMashiach kodem le-Par HaEdah b'chol ma'asav (the bull of the anointed priest precedes the community's bull in all its details). The par of the kohen mashiach goes before the par he'elem davar in all of its details. The entire service of the par kohen mashiach goes first. The kohen mashiach (anointed priest) is somebody who provides a tremendous amount of kapara (atonement) for the nation, so therefore it's fitting that his offering should go before everybody else's.