Passover, or Pesach is an important biblically derived Jewish festival. The Jewish people celebrate Passover as a commemoration of their liberation over 3,300 years ago by God from slavery in ancient Egypt that was ruled by the Pharaohs, and their freedom as a nation under the leadership of Moses. It commemorates the story of the Exodus as described in the Hebrew Bible especially in the Book of Exodus, in which the Israelites were freed from slavery in Egypt. Passover commences on the 15th of the Hebrew month of Nisan and lasts for either seven days (in Israel) or eight days (in the diaspora).
Shavuot commemorates the anniversary of the day God gave the Torah to the entire nation of Israel assembled at Mount Sinai, although the association between the giving of the Torah (Matan Torah) and Shavuot is not explicit in the Biblical text. The holiday is one of the Shalosh Regalim, the three Biblical pilgrimage festivals. It marks the conclusion of the Counting of the Omer, and its date is directly linked to that of Passover.
Sukkot, Succot or Sukkos (Feast of Booths, Feast of Tabernacles) is a biblical Jewish holiday celebrated on the 15th day of the month of Tishrei (varies from late September to late October). It is one of the three biblically mandated festivals Shalosh regalim on which Hebrews were commanded to make a pilgrimage to the Temple in Jerusalem. The holiday lasts seven days (eight in the diaspora).
Shemini Atzeret (“Eighth [day of] Assembly”) is a Jewish holiday celebrated on the 22nd day of the Hebrew month of Tishrei in the Land of Israel, and on the 22nd and 23rd outside the Land, usually coinciding with late September and/or early October. It directly follows the Jewish festival of Sukkot which is celebrated for seven days, and thus Shemini Atzeret is literally the eighth day; but it is a separate yet connected holy day devoted to the spiritual aspects of the festival of Sukkot. Part of its duality as a holy day is that it is simultaneously considered to be and celebrated as being both connected to Sukkot, yet also being a separate festival in its own right.