Hallel consists of six Psalms (113-118), which are said as a unit, on joyous occasions, when the praise of G-d is particularly appropriate, immediately following the Shacharit "Shemoneh Esray." These occasions include the following:
The three "major" festivals, Pesach, Shavuot and Sukkot, the term "major" meaning that the festival is mentioned in the "Chamishei Chumshei Torah," the Five Books of Moses
The "minor" festival, Chanukah, the term "minor" meaning that the festival is not mentioned in the Five Books of Moses, and was instituted at a later time in Jewish History, either by the members of the "Anshei K'nesset Hagedolah," the "Men of the Great Assembly," or by great Jewish leaders acting under the influence of the Divine Spirit
Rosh Chodesh (Beginnings of New Months).
These chapters are expressions of joy and faith in G-d, and of gratitude for salvation from our enemies. They were incorporated into the Book of Psalms by King David, and they were singled out for inclusion in Hallel because they contain the following fundamental themes of the faith of Judaism: the Exodus, the Giving of the Torah by G-d at Sinai, the future Resurrection of the Dead, and the Coming of the Mashiach.
On Rosh Hashanah and Yom Kippur, Hallel is not said at all, because as the Talmud explains in Masechet Arachin 10b, "Is it seemly for the king to be sitting on His Throne of Judgment, with the Books of Life and Death open before Him, and for the people to sing joyful praises to him?"
On the various Festivals, Hallel is said in one of two forms: Full Hallel and Partial Hallel.
Full Hallel, that is, all six Psalms, in their entirety, is recited on all nine days of Sukkot (including the "eighth" and "ninth" days, Shmini Atzeret and Simchat Torah), on Shavuot, on the first two days of Pesach, and on Chanukah.
Partial Hallel, which does not include verses 1-11 of Psalm 115, nor those verses from Psalm 116, is recited on the last six days of Pesach and on Rosh Chodesh. See immediately below for explanation of two day - six day split on Pesach.
Pesach, like Sukkot, has the structure of Main Holiday (two days, one in Israel), followed by Intermediate Days (four days, five in Israel), followed by Main Holiday (again, two days, one in Israel). The last two days of Main Holiday (the Seventh day, in Israel) are specifically related to the Miracle of the Crossing of the Sea of Reeds, in which the entire Egyptian army was drowned. G-d Himself declared a limitation on our expression of His praise at that time, when He said "My creatures are drowning in the sea; it is not a time for the full expression of joy." Because the Intermediate Days should not be more joyous than the Main Holiday, it was decided that only Partial Hallel would be recited on all of the last six days of Pesach.
Only the Partial Hallel is recited on Rosh Chodesh, because it was introduced at a much later time than the other Rabbinic festivals; that is, in Babylonia, in late Talmudic times. Therefore, it originally had the character only of a "minhag," or custom.
No Hallel, neither Full nor Partial, is recited on Purim, despite the fact that there occurred at that time a miraculous salvation, albeit by a "hidden" miracle, aided by the valiant actions of Mordechai and Esther, from a premeditated full-scale attack on the Jewish People, by Haman and his cohorts, for several reasons. Two of those reasons are the following:
The miracle did not occur in the Land of Israel and, for "minor" holidays, only those occurring in Israel merit the recitation of Hallel.
Even after the Miracle of Purim, the Jews remained subjects of the Persian Empire, whereas on Chanukah, as a result of the victory of the Chashmonaim, the Jews regained their independence from Greece, at least for a short while (historically speaking), before they would fall again, this time under the domination of Rome.