In the last few Mishnayot, we spoke about the rules governing which korbanot (sacrifice) are offered on the Mizbeach (altar) first. Now we'll talk about which korbanot should be eaten first. The Mishnah says: Just as the korbanot that are holier come first when they are brought as a sacrifice on the Mizbeach, so too does the same order apply to eating them. The hatat (sin offering) goes before the asham (guilt offering), and the asham goes before the todah (thanksgiving offering), and so on down that list. We don't bring up the korban olah (burnt offering), because the korban olah is entirely burned on the Mizbeach.
If there was meat from a korban shelamim (peace offering) that was brought yesterday, and meat from a korban shelamim that was brought today, shelamim could mean the sacrifice of yesterday is eaten first because we are on the clock. We have a deadline to finish it sooner. A korban shelamim is allowed to be eaten on the day it's offered, the night afterward, and then the following day. So yesterday's shelamim has to be eaten by sundown today, and the korban shelamim that we brought today, we have until tomorrow at sundown to eat it. Therefore, since we have a deadline, we eat yesterday's shelamim first so that it can be consumed within the proper time.
What if there's meat from a korban shelamim that was brought yesterday, and meat from a korban hatat or an asham that was brought today? Which one do we eat first? The korban hatat and the asham are holier, but the time for the shelamim is shorter. So when you're in conflict like this, what do you do? The Mishnah has a Machloket (dispute). “Shelamim shel emesh kodmin,” divrei Rabbi Meir: according to the Rabbi Meir, yesterday's shelamim comes first because of the deadline. We have to eat the shelamim sooner, so we would eat that first, even though it's on a lower level than the hatat and the asham. But the Chachamim (the rabbis) disagree and say that the korban hatat or the korban asham from today comes first, because they are considered to be a holier korban, and the shelamim is a less holy korban.