In the Jewish calendar, the day begins with the evening. This is true not only for the holidays, but for any calendar date. For example, a baby born in the evening has a different birth-date than a baby born before sundown. Why is this so?
In describing the order of Creation, the Torah says "There was darkness on the face of the deep; and God said 'Let there be light' — and there was light."
Clearly, the darkness existed before the light. Similarly, the description of each day ends with the phrase "and it was evening, and it was morning..." Again, the evening precedes the morning. Thus, the Jewish day begins in the evening.
This symbolizes classic Jewish optimism: If it's dark in the beginning, don't despair! It will end in great light!