The Shabbos before Purim is called Parshas Zachor (the Parshah of Remembering). Every Jew is obligated to remember that Amelek attempted to destroy the Jewish people. It is essential to remember this during the time that is in close proximity to Purim, which involved Haman who descended from Amelek trying to destroy our nation (Mishneh Berurah 685:1). There is a machlokes whether hearing Parshas Zachor is a chiyuv me'doraisa (Biblical commandment) or me'derabban (Rabbinical obligation) (Shulchan Aruch 685:7).
There is a machlokes if a person needs to hear each word or it suffices to hear the ikar ha'inyan (main idea of zichiras Amalek) in order to fulfill his obligation. Reb Shlomo Zalman Auerbach, zt"l, [Halichos Shlomo Moadim 1:18:2] and Reb Moshe Feinstein, zt"l, [Shmaitza D'Moshe 685:6] hold the ikar ha'inyan is sufficient. Reb Tzvi Pesach Frank, zt"l, holds one needs to hear each word (quoted in Sefer Shloshim Yom Kodem HaChag pg. 287 ftnt. 10). Therefore, ideally one should hear and understand each word. If he did not, he is nevertheless yotzei (Emes L'Yaakov OC ftnt. 597, Koveitz Halachos 1:8 & 9).
If one missed hearing the laining of Parshas Zachor, there is a machlokes if one can fulfill his obligation with hearing the laining on Purim morning [the Torah reading also mentions Amalek]. The Magen Avraham [685:1] holds that one is yotzei, however the Mishneh Berurah [685:16] & Aruch Hashulchan [685:5] argue. Reb Shmuel Kamenetsky, shlit"a,paskins like the MG"A (Koveitz Halachos 1:2). Therefore, if someone missed laining, it is preferable to fulfill one's obligation by hearing the laining on Purim morning, than to be yotzei on Shabbos Parshas Zachor as part of an additional laining which is exclusively for women (Koveitz Halachos 1:3).
The baal korei (person reading the Torah) needs to have in mind that he is fulfilling everyone who is listening's obligation, and each person needs to be have in mind to be yotzei (Mishneh Berurah 685:14). It is important to note, that if he is being yotzei on Purim morning with the laining, he should tell the baal koreh beforehand to have him in mind [to be motzei him] (Koveitz Halachos 1:6).
There is a machlokes whether the proper pronunciation is "Zay'cher Amalek" or "Ze'cher Amalek". Therefore the minhag is to read that Posuk two times, each time with one of these pronunciations, in order to be yotzei according to each opinion (Mishneh Berurah 685:18).
Reb Moshe Feinstein zt"l holds that one may be yotzei, even by hearing a different havara (pronunciation) that what his mesorah is. For example, an Ashkenazi may hear a Sefardi pronunciation and vice versa (Igros Moshe OC 4:23). However, Reb Shmuel Kamenetsky, shlit"a, disagrees (Koveitz Halachos 1:14).
There is a machlokes whether women are obligated to hear Parshas Zachor (see Sefer HaChinuch 603 who holds they are not obligated, however the Minchas Chinuch ibid argues). Reb Shmuel Kamenetsky, shlit"a, holds that women are not obligated; however since the customis that many women do go to shul, women should go in order not to be poraish min hatzibbur (separate oneself from the group) (Koveitz Halachos 1:10).