Popular Western presentation of Chinese astrology reduces it to a birth-year animal, which is roughly like reducing Western astrology to a Sun sign, and misses what practitioners actually do.
The working method is Bazi, the eight characters, also called the Four Pillars of Destiny. Each of the year, month, day and hour of birth is expressed as a pair, one of ten Heavenly Stems and one of twelve Earthly Branches, giving eight characters in total. The twelve branches are the familiar animals; the ten stems are the five agents in yin and yang polarity. Stems and branches run together in a repeating cycle of sixty combinations, which is the basic unit of Chinese calendrical reckoning and is attested in use for well over two thousand years.
The five agents, Wood, Fire, Earth, Metal, Water, are the analytical core. They stand in fixed relations: each generates the next in one sequence and controls another in a second, so wood feeds fire and water quenches it. The reader assesses the balance of agents across the eight characters, identifies which the day stem most needs or has in excess, and reads the chart accordingly. Luck cycles of ten years are then derived from the month pillar and read against the natal balance for timing.
A separate and equally serious system, Zi Wei Dou Shu, the purple star method, places named stars into twelve palaces and is read quite differently.
Two practical points matter. The reckoning uses the solar terms, so the astrological year begins in early February at the start of spring rather than at lunar new year, and a person born in late January belongs to the previous year's pillar. And the hour pillar covers a two-hour double hour, so the birth time is required but need not be exact to the minute.