1. At the end of davening we recite the prayer of Aleinu Leshabeach. This prayer is a great and significant one. The first paragraph of Aleinu was authored by Yehoshua Ben Nun. He recited it 7 times backwards and forwards while circling Yericho and it was one of the factors that caused the walls of Yericho to become swallowed up in the ground. The second paragraph (V’al Kein) was authored by Achan after he repented for taking the booty from Yericho. (Chida Machzik Bracha 132:2, Seder Hayom Aleinu and Meam Loez Yehoshua 6:15)
2. The Chida writes that one should recite the prayer with a tremendous amount of concentration as “there is no greater praise to Hashem [as that of Aleinu]”.
3. Harav Yaakov Emden zt”l (Siddur) adds that reciting it with kavana is a segulah as a protection “that all tzaros not befall you”.
4. Harav Yitzchak Yosef shlit”a (Yalkut Yosef Dinim L’isha page 38) writes that women are not obligated to recite Aleinu but it is a praiseworthy thing for them to do so. Similarly, Harav Pesach Falk shlit”a explains that women are technically exempt from reciting Aleinu, however, it is the correct thing for them to recite it. (Mechaze Eliyahu 20 and Halichos Bas Yisroel 2:13)
5. If one enters a shul and he hears a congregation reciting Aleinu, he should recite it with them even if he is not praying with them (the rest of davening). (M.B. 65:9)
Here is a link to a ten minute shiur that I gave on the Tefillah of Aleinu: