The etrog is widely used during the 7 day (8 day festival outside of Israel) festival of Sukkot, also called סֻכּוֹת in Hebrew. Sukkot falls on the 15th day of Tishrei, the seventh month in the Hebrew calendar, 5 days after Yom Kippur. The festival usually starts on the first few weeks of autumn. The ceremony where the etrog is used does not take place on a Sabbath during Sukkot, however.
Etrogim are used during Sukkot when Jews wave four different types of plant species in a special ceremony. G-d commands the Jewish people to do so, and this commandment can be seen in Leviticus. “On the first day, you will take for yourselves a fruit of a beautiful tree, palm branches, twigs of a braided tree and brook willows, and you will rejoice before the L-RD your G-d for seven days.” -Leviticus 23:40. The first species mentioned, “a fruit from a beautiful tree”, refers to our favorite fruit, the etrog!
The etrog and the other three species are shaken in six different directions. Jews wave the lulav (which is also the name for the bundle of the three species) and the etrog in six different directions to show that G-d is everywhere. The three species besides the etrog, in Hebrew, are the lulav, hadass, and aravah, which in English, are the leaves and branches of the date palm tree (leaf), myrtle tree (branch), and willow tree (branch) respectively.
The etrog and the other three species each have two different sets of symbolism. Each species resembles an organ/part of the body, and when Jews bring all four of them together and use them to commit a religious act, it shows that Jews will dedicate their existence to G-d. The lulav represents the spine, the hadass represents the eye, the aravah represents the mouth and the etrog represents the heart. The other set of symbolism represents four types of Jews and how they serve G-d. The lulav has a flavor but it doesn't have an aroma, so it symbolizes Jews who study Torah (the Jewish law), but they do not perform good deeds. The hadass has a pleasant aroma, but it doesn't have a taste, so it symbolizes Jews who do not study Torah but perform good deeds. The aravah doesn't have an aroma or a taste, so it symbolizes Jews who do not study Torah and who do not perform good deeds. Last, the etrog has a very pleasant aroma and a wonderful taste, so it represents Jews who study Torah and perform good deeds.
From top right to bottom left: Etrog, Myrtle, Date Palm, Willow