When the Temple stood, people would recite Hallel while offering the korban Pesaḥ, and again while eating it at night (Pesaḥim 95a). The main reason for reciting Hallel on the first night of Pesaḥ is to sing God’s praises; every Jew must see himself, on Pesaḥ night, as though he left Egypt personally, and it is only natural to sing praises to God for redeeming us. In fact, this recitation of Hallel is unique: on all other holidays we recite Hallel as an expression of praise and thanksgiving to God, but on the night of Pesaḥ we proclaim it as a song (ibid. 95b).
The Sages ordained reciting half of Hallel before the meal and half of it after the meal, so that it encompasses the eating of the korban Pesaḥ. Although we no longer have the privilege of eating the korban Pesaḥ nowadays, we eat matza instead (Maharal, Gevurot Hashem, end of ch. 62). In addition, the first half of Hallel contains Psalm 114, “When Israel left Egypt” (“Be-tzeit Yisrael Mi-Mitzrayim”), which is a continuation of the Hagada’s story. This is why, at its conclusion, we recite the blessing over the redemption from Egypt (“Birkat Ha-ge’ula”). The second half of Hallel, recited after the meal, is a more general song of thanks for all redemptions, past and future (Levush).
Another reason for dividing Hallel is that this enables us to drink all four cups over song. We drink the first cup over kiddush, the second over the first half of Hallel, the third over Birkat Ha-mazon, and the fourth over the second half of Hallel (Manhig §90).
The Rishonim are divided over whether or not a berakha should be recited over Hallel on Pesaḥ night: Some say two berakhot should be recited, one over each half of Hallel. Others say one berakha should be recited. There are also differing opinions over the wording of the berakha: some say it should be “likro et ha-Hallel” (“to recite Hallel”) and others say “ligmor et ha-Hallel” (“to complete the Hallel”). Another group of authorities maintains that no blessing at all should be pronounced over Hallel on the Seder night, either because it is divided into two parts (Rosh), because a berakha was already pronounced over the Hallel that was recited in the synagogue during the Ma’ariv prayer (Rashba), or because this Hallel is like a song and therefore requires no berakha (R. Hai Gaon). Some maintain that Birkat Ha-ge’ula covers Hallel as well. In practice, the custom is to refrain from making a berakha over the Hallel we recite at the Seder.
During the rest of the year, we stand while reciting Hallel, because it is like attesting to God’s greatness, and testimony must be given while standing. But the Sages did not wish to burden us on the Seder night, because all of our actions on this night must demonstrate freedom (Beit Yosef OḤ 422:7). Nevertheless, as we have learned, the Hagada should not be read while reclining, but with an air of solemnity (Shlah). Hallel is customarily read aloud and with sweet singing (Kaf Ha-ḥayim 480:3).