According to many opinions, reciting Hallel on various occasions is one of the seven mitzvot instituted by the Sages. Others maintain that its obligation is from the Torah or originates with the Prophets.
On eighteen days of the year (in Eretz Yisrael), the complete Hallel is recited: the eight days of Sukkot, the eight days of Chanukah, the first day of Pesach, and Shavuot. The days on which the complete Hallel is recited are called a Moed and on which work is prohibited, as the verse states: “This song shall be to you as the night of the sanctification of the festival.” (Isaiah 30:29). Although work is permitted on Chanukah, the complete Hallel is recited because of the miracle that occurred then.
On Rosh Chodesh, when work is permitted, there is no actual mitzvah to recite Hallel. Nevertheless, there is an ancient custom to recite Hallel communally, either as a remembrance of the sanctification of the new month or to publicize that it is Rosh Chodesh. To emphasize that it is only a custom, Hallel is recited with omissions. On Chol HaMoed Pesach, only Half Hallel is recited because, unlike Sukkot, a different set of sacrifices was not offered on each day of Chol HaMoed. Additional reasons for this practice have also been given. (adapted from Sichat HaShevua)