There are hundreds of melodies to Adon Olam, but this one is (at least in Nusach Anglia) *the* Rosh Hashana/Yom Kippur tune. The authorship is unknown — it is listed as Traditional in the Blue Book, and given its age, I was surprised to learn that not everyone sings (or even knows) this tune. There is a bittersweet feel to the melody as it switches between a major, low section and a minor, high section. The Blue Book specifies it should be sung slowly, though in my personal experience by the time the end of Mussaf rolls around, most of us end up singing it at a significantly faster lick! Many congregations also sing this at the end of Kol Nidrei following Yigdal.