In the musaf amidah on Rosh Hashanah and Yom Kippur, we say the Oleinu prayer. This is also said towards the end of most services when there is a custom to bow when saying וַאֲנַחְנוּ כֹּרעִים ומִשְׁתַּחֲוִים .
In the repetition of the amidah on Rosh Hashana and Yom Kippur, the chazzan, and in some places all of the worshippers, will get down on their knees at those words.
From my experience of davening in a few shuls for Rosh Hashanah and Yom Kippur (I live in Boro Park, which is predominantly hossidim) it is not done during the silent 3amidho (עמידה), but only during the repetition. Also, during the repetition, it depends on the minhag of the shul. I davened at a few places where everyone falls coree3m (כורעים) on Rosh Hashanah, some places have only the shalia7 9ibbur (שליח ציבור) fall coree3m on Rosh Hashanah, while others don't have anyone fall cor3eem on Rosh Hashanah. However, all fall on Yom Kippur. I personally fall cor3eem (during the 3amidho) everyday and I don't say 3oleinu (עלינו) during the week just on Rosh Hashanah and Yom Kippur, but I remember seeing the Tur or the Taz saying that for 3oleinu during the week it is permissible to fall cor3eem.
In the shuls that I have gone to on Rosh Hashana (Ashkenaz US) it appears that people bow in the same way as they do during the aleinu at the end of services. This is from what I was taught many years ago and observation in shul.