The Chinese zodiac runs on a twelve year cycle, and your animal is decided by the lunar new year rather than by January 1. If you were born in January or early February, you may belong to the previous year's animal. Enter your birth date and the boundary is handled for you.
The animal year changes on Chinese New Year, not on 1 January. The New Year date is not taken from a fixed table: it is calculated with the traditional lunisolar calendar rules, which follow new moons and solar terms, leap months included.