The Chachamim established that a Kohen is given the honor of the first aliyah, a Levi the second, and a Yisrael the third. The reason for this enactment is “in the interests of peace,” so there will not be any fights concerning the honor of the first aliyah. Originally, this establishment was only for Shabbat, for many people come to synagogue then, and there is more concern that tension will develop surrounding the aliyot on Shabbat (Gittin 59b). Nevertheless, the Rishonim write to practice this way on Mondays and Thursdays as well, and so it is ruled as halachah (Shulchan Aruch 135:3).
If the Kohen is equal to the Yisrael in status, even without the enactment of the Chachamim he would have to be called up before the Yisrael, for it is written concerning a Kohen, “V’Kidashto” (“You shall sanctify him”) (Leviticus 21:8). Still, Chazal’s ruling comes to establish that even if the Yisrael is greater in Torah than the Kohen, the Kohen is called up first for the sake of peace. However, if the Kohen is an am ha’aretz (uneducated person) and the Yisrael is a talmid chacham (Torah scholar), the Rishonim disagree as to the law in this case. According to the Rashba, the Yisrael is to be called up for the first aliyah, since he is a talmid chacham. However, according to Rav Amram Gaon, Rav Natrunai Gaon, and a number of other Rishonim, even if the Kohen is an uneducated person, concerning the matter of the ascent to the Torah, he should be called up even before the Yisrael who is a talmid chacham, and that is how we practice (Shulchan Aruch 135:4).
Sometimes, a great need arises to add another aliyah, such as on a Monday when two chatanim (grooms) who are both Yisraelim come to pray at the same synagogue. Since the first and second aliyot are reserved for the Kohen and Levi, if another aliyah is not added, one of the chatanim is deprived of the honor of being called up to the Torah. Although according to the Rama it is permitted to add an aliyah for this reason, in practice it has been ruled that it is forbidden to add to the already existing three aliyot (Shulchan Aruch 135:1; Mishnah Berurah 3). The advice given is to ask the Kohen to leave the synagogue at the time of the first aliyah. Then, when no Kohen is present, a Yisrael will be called up for the first aliyah, thereby allowing both chatanim to be called up to the Torah that day (see Yabia Omer, part 6, 23).
