Rosh Hashanah is the first in a series of very important Jewish festivals. The name translates as "Head of the Year", and is seen as the beginning of the year for the judgment of mankind by G-d. The cycle of Torah readings does not return to the start of Genesis until Simchat Torah, and the festive cycle begins each year with the first of the month of Nisan.
Coming ten days before Yom Kippur, the Day of Atonement, it begins a period of contemplation and study culminating in the day of prayer and fasting - the Ten Days of Repentance. During these ten days, we traditionally apologise to the people we have wronged over the year, and ask to be forgiven. Only when we are forgiven by the people we hurt can we possible be forgiven by G-d.
On Rosh Hashanah, it is customary to dip apple into honey, bless it, and eat it. This is to symbolise what we hope will be a sweet new year. On the afternoon of the first day (unless it is Shabbat), some people like to cleanse their sins through a ritual known as Tashlich. They immerse themselves in a river or spring, to literally cleanse themselves. Another tradition around tashlich involves throwing bread crumbs into the water, or emptying out your pockets into the water.