Sukkot runs seven days. Then comes a day the Torah calls the eighth, which is not the eighth day of Sukkot.
Everything that defines Sukkot stops. The sukkah is no longer required and the four species are put away. The festival offerings, which had descended day by day through Sukkot from thirteen bulls to seven, become a single bull. The rabbis treat it as a separate holy day with its own sacrifice, its own name and its own rules, and the Talmud puts the question directly, in what respect is it distinct?, before answering with a list of technical particulars.
The interpretation most often given is affectionate rather than legal: the guests are leaving after a week, and one more day is asked of them. Rashi records it in roughly those terms.
Its liturgical work is the turn of the season. The prayer for rain is said for the first time, and from this day the daily prayers acknowledge wind and rain until Passover. In a calendar tied to the agricultural year of the land of Israel, that is the moment the year pivots.
And Simchat Torah
Simchat Torah, finishing Deuteronomy, immediately beginning Genesis, and dancing seven circuits with the scrolls, is not biblical and is centuries younger. It grew out of the Babylonian practice of reading the whole Torah in one year rather than the three used in the land of Israel, and it attached itself to a day already sitting on the calendar with no observance of its own.
Where it falls now depends on where you are and who you ask. In Israel and in Reform congregations there is one day, Tishrei 22, carrying both. In Orthodox and Conservative communities in the diaspora the second festival day, Tishrei 23, takes Simchat Torah, leaving Shemini Atzeret to itself.