The Chatzi Kaddish prayer is among the best-known sections of the services of Rosh Hashanah and Yom Kippur. It is recited before the Musaf (additional) service. On the Jewish High Holidays, it is sung to a stirring melody by the hazzan (a/k/a cantor, or prayer-leader). The music is a staple of cantorial singing. Indeed, it is one of the beloved pieces of Jewish Music or of a Jewish song. Chazzan Marc Berman chants the prayer in this video.
Among Jewish songs of praise and worship, this Kaddish is one of the most venerable, comparable to , say, "Kol Nidre," "Aleinu" or "U'netaneh Tokef." The Chatzi Kaddish is found in both the Rosh Hashanah prayers and in the Yom Kippur prayers. That is, it is recited at both Rosh Hashanah services and in the Yom Kippur services.
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"Chatzi Kaddish" (or "Khatzi" Kaddish*) literally means "half-Kaddish." It is an abridged version of the full Kaddish recited near the end of every service. This half-kaddish, in contrast, generally marks the formal beginning of the Musaf service, which follows the morning service and the Biblical readings of the day. The prayer extols the Deity and expresses the hope that He will soon be universally recognized.
This Chatzi Kaddish on the Yamim Noraim (Jewish high holidays) is also sometimes referred to as the "great" Kaddish. (Not to be confused with the long "great kaddish" ("Kaddish Hagadol") recited in the burial service at a cemetery.)
The musical version sung here is the traditional Ashkenazi (Eastern European) nusach. "Nusach" refers to the traditional musical prayer mode. Each holiday has its own nusach. Indeed, there is usually a different nusach for each section of each service on any given holiday.
"Nusach" can also refer to the text of a service, which varies slightly between different rites that emanate from different geographical locations.

*Pronunciation Note: The "ch" (or, sometimes, "kh" or just "h") in "Chatzi, "nusach" and "chazzan" is pronounced as a voiceless uvular fricative trill [(x) in the International Phonetic Alphabet ]. This sound is common in Hebrew, but does not exist in some languages, including English. One can hear it several times in the recording.