On this date, Rabbi Schneur Zalman of Liadi, who supported and aided the Czar's army during the Napoleonic wars, was forced to flee his hometown from Napoleon's forces which were advancing through White Russia in their push toward Moscow. After five months of wanderings he arrived in the town of Pyena. There he fell ill and, weakened by the tribulations of his flight and the harsh Russian winter, passed away on the 24th of Tevet, 5573 (1812).
Links: The Rebbe vs Napoleon
R. Eliezer Zusia Portugal (1898–1982), the Skulener Rebbe, is best known for his superhuman efforts to rescue Jewish orphans and refugees in Eastern Europe before, during, and after World War II and his continuing support of them. He immigrated to the United States in 1960, after imprisonment in Rumania and international efforts to secure his release. He was renowned for his lengthy prayers and the beautiful tunes he composed.
This Shabbat is Shabbat Mevarchim (“the Shabbat that blesses" the new month): a special prayer is recited blessing the Rosh Chodesh ("Head of the Month") of the upcoming month of Elul, which falls on Sunday and Monday of next week.
As noted a here, this day is especially significant since this is the final (and culminating) such blessing of the year.
Prior to the blessing, we announce the precise time of the molad, the "birth" of the new moon. See molad times.
It is a Chabad custom to recite the entire book of Psalms before morning prayers, and to conduct farbrengens (chassidic gatherings) in the course of the Shabbat.
Links: Shabbat Mevarchim; Tehillim (the Book of Psalms); The Farbrengen
During the summer months, from the Shabbat after Passover until the Shabbat before Rosh Hashahah, we study a weekly chapter of the Talmud's Ethics of the Fathers ("Avot") each Shabbat afternoon; this week we study Chapter Five.