The Tosefta (Chapter 3 of Sukkah) states: “There are eighteen days and one night throughout the year when the (complete) Hallel is recited, as follows: The eight days of the Sukkot holiday, the eight days of Chanukah, the first day of Pesach as well as the first night of Pesach, and on the holiday of Shavuot.” Our Sages in Masechet Sofrim (Chapter 20, Halacha 9) states: “It is especially worthy to recite the Hallel pleasantly in order to fulfill the verse, ‘And let us exalt His name together.’”
The above serves as the source for Sephardic Jews and the Jews of Israel who customarily recite the complete Hallel with its blessings on the first night of Pesach following the Arvit Amida prayer. Indeed, the Tur (Chapter 473) states: “How good and pleasant is the custom of reciting the Hallel along with the congregation in the synagogue on the first night of Pesach with its blessings; there is indeed a source for this custom in Masechet Sofrim.”
Clearly, reciting Hallel on the night of Pesach is because of the miracle of the exodus from Egypt which is the time when Hashem delivered us from slavery to freedom, as the Talmud Yerushalmi (Pesachim, Chapter 5, Halacha 5): “Rabbi Levi said: On that night, Hashem made Pharaoh’s voice resound throughout the entire land of Egypt and he said, ‘Get up and leave from the midst of my nation! Until now, you were the slaves of Pharaoh. From this point on, you are now Hashem’s slaves!” At that moment, the Jewish nation began to praise Hashem and said, “Praise Hashem! Praise, oh servants of Hashem and not the servants of Pharaoh.” Indeed, in every generation, one must envision as though he himself has left Egypt.
Another reason for reciting the Hallel on the night of Pesach is because when the Jewish nation was in Egypt, they recited the Hallel while slaughtering the Pesach offering. This custom quoted by the great Acharonim and Mekubalim who speak lengthily about the virtues of reciting the Hallel on the night of Pesach before Kiddush. Several Ashkenazi luminaries observed this custom as well, including the great Noda Bi’huda (Hagaon Rabbeinu Yechezkel Ha’Levi Landau, head of the rabbinical court in Prague) who would recite the Hallel following Arvit prayers although the custom of the people of that city was not to recite Hallel on the night of Pesach in accordance with the ruling of the Rama. (See Teshuva Me’Ahava, Chapter 90)
Regarding all other holidays, women do not recite a blessing before and after reciting the Hallel, for Hallel is considered positive, time-bound Mitzvah which women are exempt from performing based on the words of the Tosafot (Sukkah 38a). Nevertheless, on the first night of Pesach when women are obligated in all of the Mitzvot of the Seder night in the same manner as men, Maran Rabbeinu Ovadia Yosef zt”l writes (in his Responsa Yechave Da’at, Volume 5, Chapter 34) that women are likewise obligated to recite the complete Hallel along with its blessings before Kiddush, i.e. before the Seder begins. This is actually the only time of year that Sephardic women may recite a blessing on the Hallel and they must recite it completely along with its blessings, beginning and end, since they were also included in this miracle and they are likewise obligated in all of the Mitzvot of the Seder night. Indeed, in the merit of righteous women, we were redeemed from the bondage of Egypt and in the merit of righteous women shall we merit the Ultimate Redemption!