The ten days starting with Rosh Hashanah and ending with Yom Kippur are commonly known as the Days of Awe (Yamim Noraim) or the Days of Repentance. This is a time for serious introspection, a time to consider the sins of the previous year and repent before Yom Kippur.
One of the ongoing themes of the Days of Awe is the concept that God has “books” that he writes our names in, writing down who will live and who will die, who will have a good life and who will have a bad life, for the next year. These books are written in on Rosh Hashanah, but our actions during the Days of Awe can alter God’s decree. The actions that change the decree are “teshuvah, tefilah and tzedakah,” repentance, prayer, good deeds (usually, charity). These “books” are sealed on Yom Kippur. This concept of writing in books is the source of the common greeting during this time is “May you be inscribed and sealed for a good year.”
Among the customs of this time, it is common to seek reconciliation with people you may have wronged during the course of the year. The Talmud maintains that Yom Kippur atones only for sins between man and God. To atone for sins against another person, you must first seek reconciliation with that person, righting the wrongs you committed against them if possible.
Work is permitted as usual during the intermediate Days of Awe, from Tishri 3 to Tishri 9, except of course for Shabbat during that week.