Al HaNissim (Hebrew: עַל הַנִסִּים, "on the miracles") is an addition to the Amidah and Birkat Hamazon on Hanukkah and Purim. On both holidays, it starts off with a short paragraph, beginning with the words for which it is named. After that, each holiday has a unique paragraph, describing the events for which that day is celebrated.
On Hanukkah, the prayer is recited to thank God for the miracles with which the holiday is associated. On Purim, it is recited because there is an obligation to recite the events of the day in one's prayers, but not the miracles of the day.
Regarding "Al Hanisim" on Yom Ha'atzmaut
Various rabbis endorsed the recitation of Al Hanisim on Yom Ha'atzmaut, and even penned unique versions of this prayer. The first to publish a version of Al Hanisim for Yom Ha'atzmaut was Rabbi Ezra Zion Melamed.<ref name=Yom Ha'atzmaut>see more about this in the parallel entry in Hebrew wikipedia & wikisource. See also: Halakhic Permissibility of the recitation of Al Hanisim on Yom Haatzmaut</ref>