Just as Jewish farmers are commanded to offer tithes to the kohanim from their produce, similarly sections of animals are to be given to them, as well. According to the Torah (Devarim 18:3), whenever an animal is slaughtered for food, the owner must give to the kohen three sections of the animal - the zero'ah (shoulder), lehayayim (cheeks) and kevah (stomach).
Although the Torah is clear with regard to the types of animals involved in this mitzvah, as well as what must be given and to whom it is given, nevertheless there are many questions that need to be clarified. These issues are the focus of the tenth perek (=chapter) of Masechet Hullin, which begins on today's daf (=page).
The first Mishnah opens by teaching that the requirement to give the zero'ah, lehayayim and kevah to the kohen applies in both Israel and the Diaspora, whether or not the Temple is standing and is limited only to ordinary animals, not to sanctified ones.
In explanation for why these particular parts of the animal were chosen to be given to the kohanim, the commentaries offer a number of possibilities. One approach is that these are the most preferred pieces of the animal - the shoulder in the body of the animal, the tongue in the head of the animal and the stomach as part of the animal's innards. Furthermore, they come as a reward for the sacrificial service performed by the kohanim: The kohen slaughters the sacrifice (represented by the zero'ah), recites a blessing (lehayayim) and finally butchers the animal for sacrifice, checking its innards for any possible blemishes (kevah). Finally, the Gemara (daf 134b) suggests that this rewards the kohanim for the act of Pinhas ha-kohen who put a stop to a public act of idolatrous harlotry by killing Zimri and Cozbi (see Bamidbar Chapter 25) by arming himself with a spear (represented by the zero'ah), stabbing them (kevah) and praying (lehayayim).