The thirteenth section of the Seder is Hallel. Hallel is the section in which we glorify and praise Hashem. The source for reciting Hallel during the Seder is the Mishna in Pesachim (9:3). The Gemara in Pesachim on 95b says that we derive this practice from the Pasuk in Yeshayahu (30:29) that implies that one must sing on the first night of a holiday. Rashi explains that this refers to Pesach.
The Tur cites a major Machloket about whether or not a Beracha needs to be recited before Hallel on the Seder night. He says that some people used to recite two Brachot, one on the part of Hallel at the end of Magid, and a second on the rest of Hallel. The Tur also quotes other Rishonim who say that you don't say any Beracha. The Tur concludes that since it is a doubt as to whether you should say a Beracha or not, the solution is not to say any Beracha at all. There is a compromise opinion who says that only one Beracha should be recited at the beginning of Hallel in Magid, because the meal is not a significant interruption. The Bach says that it is common practice not to recite a Beracha and that is why the Shulchan Aruch doesn't make a mention of a Beracha on Hallel.
The Ran quotes another opinion as to why there is no Beracha said with Hallel at the Seder. This is the opinion of the Rav Hai Gaon, who says that on the Seder night Hallel is a Shira and not a Keriah. It is sung and not read. Since it is not done in its normal way that is why there is no Beracha. This could possibly be the reason why we sit when we say Hallel at the Seder, although we stand when saying Hallel in Shul. Another explanation is that all the other recitations of Hallel take place during the day, because it is a Mitzva, so since this time it's said at night it isn't really a Mitzva, so no Beracha needs to be recited. The Orchot Chaim suggests that the paragraph of -5*,+ which is right before Hallel in Magid replaces the Beracha before Hallel.
Many of the Poskim, including the Ran say that it is a Minhag to recite Hallel in Shul on Pesach night with a Beracha. This is in order to not have to make the Beracha at the Seder. The Rashba says that Hallel was instituted on Pesach night to be said in Shul with a Beracha at shul but not at home. The Rama says that the Minhag is not to recite Hallel in Shul on Pesach night at all. The Vilna Gaon then states that originally we said Hallel with a Beracha in Shul because this way the people who couldn't do it later could fulfill it now. In another place he says that saying Hallel in Shul is a good idea because it is Pirsumei Nisa, publicizing the miracle of Hashem. In practice, many Ashkenazim do not recite Hallel at all in Shul on Pesach night, while the Sephardim and Chassidim do say Hallel in Shul with a Beracha.
Rav Chaim Soloveitchik insisted on reciting Hallel in Shul at Pesach night. The Rav explains that reciting Hallel in Shul "permits" us to recite Hallel in our home.