There is an ancient enactment, from the time of Moshe, that the Torah is read in public on Shabbat day, Monday, and Thursdays from a scroll written with ink on parchment, so that three days never pass without the study of Torah. The Sages teach that this enactment was instituted based on the verse, “They traveled for three days in the desert without finding any water” (Shemot 15:22). Moshe and his disciples, the elders and the prophets, understood that the thirst for water was a result of three consecutive days in which Israel did not engage publicly in Torah (BK 82a). Torah is likened to water, for just as water sustains all animal and plant life, so too Torah sustains the soul. When the people disconnected themselves from the Torah even slightly, water sources also ceased to flow. Although the Torah scholars of that generation presumably studied Torah during those three days, the meaning here is that for three days the people of Israel did not engage in Torah publicly. Therefore, it was established that the Torah would be read every Monday, Thursday, and Shabbat, so that never again will more than three days pass without Israel publicly reading from the Torah.
Ezra the Scribe further instituted that the Torah be read at Minĥa on Shabbat because of “yoshvei keranot” (BK 82a). Some say that yoshvei keranot are merchants and craftsmen who are unable to hear the reading on Mondays and Thursdays, and so the Shabbat Minĥa reading was added as a make-up opportunity for them. Others maintain that yoshvei keranot are frivolous individuals. There was concern that after they finish Shaĥarit on Shabbat they would turn to drink and idleness, and so Torah reading was instituted at Minĥa (see Peninei Halakha: Shabbat 5:8). The Sages further enacted that relevant Torah portions are read on all festivals and Rosh Ĥodesh.
On Mondays and Thursdays three people are called up (“olim”) to the Torah; four on Rosh Ĥodesh and Ĥol Ha-mo’ed; five on Yom Tov; six on Yom Kippur; and seven on Shabbat (Megilla 21a). Initially, it was customary that each person called up to the Torah would read that section, but as time went on, most congregations designated a Torah reader (ba’al koreh) who would read the Torah for everyone. This is so those who do not know how to read the Torah are not insulted and so the reading itself is especially accurate (Peninei Halakha: Collected Essays I 4:6; II p. 227)
Technically, it is permissible to add aliyot on Shabbat and call up more than seven people to the Torah, as long as each person called up for an aliya reads at least three verses. However, for several reasons, it is preferable not to add aliyot. Only when there is a pressing need, such as for a joyous occasion with celebrants who will be insulted if they are not called up, may aliyot be added to the standard seven (SA 282:1-2; MB 4-5; see Peninei Halakha: Shabbat 5:6).
The reading must be done from a kosher scroll, written with the proper intent, in ink, on parchment, just as the first Torah scroll was written by Moshe at God’s dictate. The reading must be in the presence of a minyan, as it is davar she-bikdusha (see above, 2:10, which states that women are exempt from Torah reading, and below, 22:4-6, regarding Torah reading on Shabbat. See also Peninei Halakha: Collected Essays I ch. 4 and Prayer ch. 22).