Adding "HaMelech HaKadosh" and "HaMelech HaMishpat" in Prayer: Berachot 12b
Repeating the Amidah if one forgets special alterations for these days: Berachot 12b, Tos. Berachot 12b #2
Closeness to Gd During These Days
These days are considered to be a time when HaShem is more readily available: Yevamot 105a
Efficacy of repentance by the Individual, and by the Community, during this time period: Rosh HaShanah 17b-18a
Ability to appeal Gd's verdict during this time period: Rosh HaShanah 18a
Other
Testing the flow of light from a flame during the days between Rosh haShanah and Yom Kippur, to see whether the year will flourish: Keritot 5b
