In the times of the Bet Ha’mikdash, the Kohanim did not receive a portion of the Land of Israel, and thus they were unable to produce food. They were supported by various gifts that the rest of the nation was required to give them.
As the Torah instructs in Parashat Shoftim, these gifts include certain portions of any kosher animal that is slaughtered – specifically, the Zeroa (arm), Lehayayim (cheeks) and Keba (stomach). This requirement applies when any ordinary, non-sacrificial animal is slaughtered. The food is not hallowed, which means that it may be eaten by anybody if the Kohen decides to share it, and it may be eaten even in a state of Tuma (impurity).
Various reasons have been suggested for why specifically these portions were chosen as gifts for the Kohen. Rashi explains that these portions commemorate the heroic act performed by Pinhas, one of the first Kohanim, who killed Zimri and Kozbi, a man and a woman who committed a public sinful act. Pinhas prayed for G-d’s help at that time, commemorated by the Lehayayim, which are in the mouth; he killed them with his arm, commemorated by the Zeroa; and he stabbed them in the stomach, commemorated by the Keba.
The Keli Yakar (Rav Shlomo Efrayim Luntschitz, 1550-1619) explains differently, suggesting that these gifts are given in exchange for the Birkat Kohanim blessing which the Kohanim confer upon the nation. The Kohanim therefore receive the Lehayayim, representing the mouth, which they use to recite the blessing, and the Zeroa, which symbolizes the Kohanim’s raising their hands as they pronounce Birkat Kohanim. In this Beracha they bless the people with prosperity and satiation, represented by the Keba, the animal’s stomach.
Rav Abraham Saba (1440-1508) adds that in general, Kohanim are given gifts of meat because they are to devote themselves to Torah study, which has the effect of weakening a person, and so they need meat to keep them healthy and strong.
If these portions are not given to a Kohen, the rest of the animal is nevertheless permissible for consumption. These gifts differ in this respect from Teruma, a portion of produce which must be separated before the rest of the produce may be eaten. This is proven from a story told by the Gemara in Masechet Megilla of, Rabbi Preda, who was once asked why he earned such a long life, and he replied that he never ate meat from an animal before these portions were given to a Kohen. It is clear from this story that waiting for these portions to be given before eating the rest constitutes a Midat Hasidut – special measure of piety, that is not required according to Halacha.
The Sefer Ha’hinuch (anonymous work from the 13th century) writes (506) that this obligation cannot be enforced, because no particular Kohen can claim rights to these portions. Since the animal’s owner is entitled to choose to which Kohen he wishes to give these portions, no Kohen can make a legal claim that he is owed these parts of the animal.
The Shulhan Aruch (Yoreh De’a 61:1) rules explicitly that the requirement to give these portions to a Kohen applies even nowadays. Later (61:21), he brings two opinions as to whether this obligation applies only in the Land of Israel, or also in the Diaspora. The Gemara in Masechet Shabbat (10b) tells that Rav Hisda, who was a Kohen and lived in Babylonia, received these gifts, which certainly implies that this obligation is binding even outside the Land of Israel. Rashi, however, commenting on this story, references the Gemara’s ruling elsewhere, in Masechet Berachot, that the accepted custom follows the opinion that Reshit Ha’gez – the obligation to give a Kohen the first shearing of a sheep’s wool – does not apply outside Eretz Yisrael. According to Rashi, this ruling applies also to the portions of a slaughtered animal, and he writes that for this reason, the accepted practice is not to give these portions to a Kohen.
Others, however, including the Rif (Rav Yishak of Fez, Morocco, 1013-1103), the Rambam (Rav Moshe Maimonides, Spain-Egypt, 1135-1204) and the Ramban (Rav Moshe Nahmanides, Spain, 1194-1270), disagree. In their view, the law regarding Reshit Ha’gez has no bearing at all on the obligation to give these portions of an animal, and thus this obligation applies both in Eretz Yisrael and in the Diaspora. This is the ruling of the Hid"a (Rav Haim Yosef David Azulai, 1724-1806), in his Mahazik Beracha (Y.D. 61:14). The Shulhan Aruch, however, after bringing both opinions, writes that the accepted custom follows the lenient position, that this obligation does not apply in the Diaspora. This appears to be the accepted practice even today. Interestingly, however, there are stories told of great Sadikim, such as the Hatam Sofer (Rav Moshe Sofer, Pressburg, 1762-1839), who made a point of fulfilling this Misva even in the Diaspora.
This explains why this Misva is not fulfilled here in the Diaspora – because the Shulhan Aruch ruled that the accepted practice follows Rashi’s view. It does not explain why this is not commonly observed in Israel. Already hundreds of years ago, Maran (author of the Shulhan Aruch) published a letter by one of his contemporaries in his work Abkat Rochel bemoaning the neglect of this Misva. (This Rabbi felt that it should be observed even in the Diaspora.) The letter states that Rav Levi ibn Habib (Jerusalem, 1480-1545) instituted a solution to this problem, implementing a system whereby butchers paid a particular sum of money to a fund for each animal slaughtered, and this fund would be distributed to Kohanim. This system is mentioned also by the Mabit (Rav Moshe of Trani, 1505-1585) and by his son, the Maharit (Rav Yosef of Trani, 1568-1639). The author of this letter, however, felt that this system was not sufficient, as in his view, the actual portions of meat must be given to a Kohen.
Rav Yechiel Michel Tuketchinsky (1871-1955) tells of a different practice that was followed – whereby a large "loan" was given to the Kohanim, and each time an animal was slaughtered, a certain sum was deducted from the amount owed by the Kohen.
It should be noted that if the animal is co-owned by a non-Jew, then this requirement does not apply. Conceivably, then, this obligation can be circumvented by granting a non-Jew a share in all the animals in the butcher shop. If so, then perhaps we might say that since many butcher shops have an arrangement whereby animals which are found to be Terefot (mortally wounded), and thus forbidden for consumption, are given to a non-Jew, every animal slaughtered is, in a sense, co-owned by a non-Jew until its status is determined. This might perhaps provide a basis for slaughterhouses that do not ensure to give these portions, or their monetary equivalent, to a Kohen. However, this question requires further elucidation.
Summary: The obligation of "Zeroa, Lehayayim Ve’keba" requires giving a Kohen certain portions of every kosher animal that is slaughtered. According to accepted custom, this requirement does not apply in the Diaspora.