You shall sanctify him, for he offers up the food offering of your God; he shall be holy to you, for I, the Lord Who sanctifies you, am holy. (Leviticus 21:8)
The Mishnah states that a Kohen (priest) takes precedence for an aliyah (call up) to the Torah over a Levi and a Levi takes precedence over a Yisrael (Horaot 13a). Yet this is only one well-known example of the honour given to Kohanim.
Similarly, the Gemara infers from this verse that we must ‘sanctify’ Kohanim today by granting them honour (Gittin 59b). According to Rabbi Yisrael Meir HaKohen (known as ‘the Cofetz Chaim’, d. 1933) we honour the Kohanim by asking them to recite Kiddush, birkat hamotzi (the blessing over bread) and lead birkat hamazon (the blessing after the meal) at communal mealtimes (Mishnah Berurah 201:13).
Nevertheless, the head of a household where a meal is being served in his house is not obligated to honour guests who are Kohanim in this way (Mishnah Berurah 135:9). If a non-Kohen does lead birkat hamazon in the presence of a Kohen, he must include the Kohen when asking for permission (after the word bir’shut) in the zimmun (invitation to bless) at the beginning of birkat hamazon even if he is the head of the household (see Chief Rabbi’s green siddur page 758).
Rabbi Moshe Isserlis (known as ‘the Rama’, d. 1572) adds that a Kohen can forego his honour except in being called up to the Torah (Rama 128:45). If there are circumstances whereby the presence of a Kohen in the synagogue would exclude someone from receiving an aliyah on a special occasion, he may leave the room before the Torah reading starts to avoid being called up. For example, if two Yisraelim are both getting married on a Thursday morning, they can both be called up by giving the first aliyah to a Levi and then receive the subsequent two consecutively.
According to Maimonides (known as the Rambam, d. 1204) since God chose the Kohanim to perform the services and offerings in the Bet Hamikdash (Temple), they are God’s emissaries. Therefore, when we honour a Kohen, by extension we honour God (Sefer Hamitzvot 32).