What is Parshat Zachor, what does it have to do with Purim, and why is it read on Shabbat and not on Purim itself? Thanks for your time.
It is a positive Torah commandment to remember and verbally recall the wicked attack perpetrated by the people of Amalek against our ancestors upon their liberation from slavery in Egypt. Since Haman and his followers were descendents of this people who intended to execute the annihilation planned by their forebears, the Talmudic sages prescribed that this remembrance take place in conjunction with Purim.
The remembrance itself is conducted by reading the relevant passages from the Torah during the communal Torah reading. Even though these verses appear in Parshat Ki Tetzei which is read in the summer, they are read separately as Parshat Zachor to fulfill the above-mentioned Torah commandment at the time decreed by the Sages – the Shabbat before Purim. The verses are:
"You shall remember what Amalek did to you on the way, when you went out of Egypt, how he happened upon you on the way and cut off all the stragglers at your rear, when you were faint and weary, and he did not fear G-d. So it will be, when the Lord your G-d grants you respite from all your enemies around you in the land which the Lord, your G-d, gives to you as an inheritance to possess, that you shall obliterate the remembrance of Amalek from beneath the Heavens. You shall not forget!" (Deut. 25:17-19).
The reason Parshat Zachor is read on the Shabbat before Purim is that Shabbat is a time when the entire community is gathered in the synagogue, which ensures that the mitzvah will be performed by the entire People of Israel. Since the mitzvah is performed by reading the relevant passage from the Torah, Shabbat was chosen as a time that the community is anyways engaged in communal Torah reading. The Shabbat before Purim was chosen in order to proclaim the upcoming holiday to the people and to prepare them for its message.
The order of the Torah reading is as follows: two Torah scrolls are taken from the Holy Ark. In one the regular weekly Torah portion is read in its entirety with seven aliyot. In the other Torah scroll, the above-quoted passages are read as maftir, the concluding aliya. Since this passage opens with the phrase, "Remember what Amalek did to you", which starts with the Hebrew word Zachor, this Shabbat is referred to as Shabbat Zachor. The Haftara selection for Shabbat Zachor (I Samuel 15) is a dramatic tale of the historical enmity between Amalek and Israel.
As mentioned earlier, it is a Torah commandment to hear the public Torah reading of Parshat Zachor. However, there is a difference of opinion as to whether women are included in this obligation. Some commentaries are of the opinion that since this reading recalls the requirement to war against Amalek's threat to G-d and the Jewish People, women, who are generally exempt from waging war, are exempt from hearing this reading. Others point out that the distinction between men and women in battle applies only to "voluntary" war, but not to "commanded" war, where even women are commanded to fight. Accordingly, since the war against Amalek is commanded, women are also required to hear Parshat Zachor. Because of the doubt, women are considered to be exempt from hearing the reading, but nevertheless make an effort to go to hear it if they can.