In the villages of Eastern Europe it was a custom before the New Year for a messenger to go from house to house with a sack. Those who could afford it put coins into the sack; those who were poor took coins from the sack. No one knew who gave and who took. No one was embarassed because he was poor. Every family had money to buy the things they neeed to celebrate the holiday.

Giving tzedakah, sharing what we have with those in need, is an important mitzvah in Jewish life.

The ideal of charity around the New Year time has its place in the modern world in the attempts of some Jewish communal service agencies and local community-based projects to make sure that all Jews can afford to attend services and get a good Yom Tov meal. But our concern for the unfortunate is not limited to the borders of our own country. It is by now something of a tradition to find an appeal from the State of Israel on one's seat in synagogue around the High Holidays.
Other traditions: Tashlich , Food

First two paragraphs by Judyth Robbins Saypol
From "My Very Own Rosh Hashanah"
Reprinted with permission