The ten days from Rosh Hashanah through Yom Kippur, during which repentance (teshuvah) is a key theme.
The ten days from rosh-hashanah-new-year through yom-kippur, during which repentance-teshuvah is a key theme. There are additions to the prayers to this effect, including changes to the Shemoneh Esrei and adding Avinu Malkeinu. Some add Psalm 130 before Barchu or Tachanun. A common custom is to learn one of the ten chapters of Mishne Torah Hilchot Teshuva on each day.