Hallel, literally 'praise', consists of Psalms 113-118, and is added to the prayers on holidays.
A "full Hallel" includes the complete 5 chapters of Tehillim, in addition to a bracha before and after. In a "half Hallel", 115:1-11 and 116:1-11 are skipped. In Sephardi communities, the brachot are not said; the Ashkenazi custom is to say them.
A whole Hallel is said on Chanukah, the first day of Pesach, Shavuot, and Sukkot. Some communities add it on Yom Ha'atzma'ut and Yom Yerushalayim. A half Hallel is said on Rosh Chodesh (except Rosh Hashanah) and the last 6 days of Pesach.