In brief, the High Holidays - also known as the High Holy Days - are Rosh HaShanah and Yom Kippur as well as the ten days that fall between them. These ten days are collectively known as the "Days of Awe" (Yamim Nora'im) or the "Ten Days of Repentance" (Aseret Yamei T'shuvah).
Rosh HaShanah is the Jewish New Year and is followed by the Days of Awe. During this time Jews are encouraged to reflect upon the past year and to atone for their sins by sincerely apologizing to anyone they may have wronged. The idea is that, by reconciling their differences with others and committing to do better in the coming year, Jews can observe Yom Kippur with a clear conscience. Also known as "The Day of Atonement," during Yom Kippur it is believed that God will determine the fate of each person for the coming year.
To learn more about the High Holidays visit: What Are the Jewish High Holidays?