Traditionally, the 9th of the Hebrew month of Av is the day of mourning and supplication and fasting and commemoration of all the suffering that the Jewish people went through over the centuries. This includes the Holocaust, of course.
There are two other nontraditional memorial days of Holocaust. The International Holocaust Remembrance Day is January 27th, when the Red Army liberated the Auschwitz-Birkenau concentration camps in 1945. The other is the 27th of the Hebrew month of Nissan, called Yom HaShoah, and is commemorated in Israel and by Jewish communities around the world in solidarity with Israel. The 27th of of the Hebrew month of Nissan (which falls either in April or May in the Solar calendar) marks the irruption of the Warsaw ghetto uprising (in April 19, 1943), which makes sense for Israel as the nation-state of the Jewish survivors of the Holocaust. (The Warsaw Ghetto uprising that remarkably fended off the German wehrmacht, undoubtedly at the time the most powerful army in the world, for over one month, irrupted after the last ghetto residents found out [through the help of the Polish underground resistance movement] that over three million of their brethren had been systematically gassed in the German-occupied Poland between the springs of 1942 and 1943.)
There is an eerie difference between the traditionalday of commemoration and the nontraditionalmemorial days. For one, both memorial days commemorate the human motive for liberation from collective suffering. The traditional day of commemorating collective suffering had been designated over time as the day of disaster, the cruel suffering itself. In other words, the most cruel suffering in Jewish history is collapsed into the one traditional day: the 9th of Av.
The 9th of Av is especially when the sacking of the 1st and the 2nd Jerusalem Temples had been commemorated in Jewish communities for over two millennia. In fact, the 9th of Av is not the day when both Temples fell. The 1st Jerusalem Temple fell in 586 BCE on the 7th of Av and the 2nd Jerusalem Temple fell in 70 CE on the 10th of Av. But both dates were collapsed into one, the 9th of Av. In the course of the last two thousand years, the most dramatic events of Jewish suffering were added to the same day of commemoration, fasting, and supplication, according to the traditional Jewish calendar, even if those events really happened on completely different days. One of the primary reasons for this conflation is rather simple: to synchronize the mourning period of the various Jewish communities scattered about the world.
The second main difference between the traditional and nontraditional days of remembering the Holocaust is the major conflict between the traditional and the non-traditional reckoning. While the International Holocaust Remembrance Day on January 27th does not seem to conflict with the accepted Jewish calendar of the last two millennia, Yom HaShoah does conflict with the month of Nissan, which is the first month of the traditional Jewish calendar, a month of joy and celebration in the birth of the Jewish people as highlighted by the prevalent communal celebration of Passover (Pesach) to this very day. Life is full of contradictions.