Chinese astrology
This is related to the Chinese calendar, particularly its 12-year cycle of animals.
The twelve zodiac animal signs are rat, ox, tiger, rabbit, dragon, snake, horse, goat, monkey, rooster, dog and pig in that order. The legend had it that the rat was given the task to invite the animals to report to the heaven to be selected for the zodiac signs. The cat was a good friend of the rat, but the rat forgot to invite him. So the cat vowed to be the rat's natural enemy for ages to come.
There is also a cycle of the five Chinese element on top of the cycle of animals. So the year sign can be a gold dragon, a wood rooster etc. In ancient match-making practice in China, couples were brought together according to their compatible signs. For example, it is believed that dog and dog don't get along, but dog and pig do; a water dog supports a wood pig but dominates a fire pig in their relationship because water benefits wood, but controls fire according to the Chinese five elements' interaction.
Since the Chinese zodiac follows the Chinese lunar calendar, so the switch over date for each sign is according to the Chinese new year, not January 1 of the western calendar. The Chinese zodiac signs are used by cultures other than Chinese also. For example, some Japanese people use them. The United States Postal Service issues a "Year of the ???" postal stamp each year to honor this Chinese heritage. However, those who don't use the Chinese lunar calendar, such as the Japanese, usually just assume the signs switch over on Jan 1 of each year.
However, those who are serious about the fortune telling aspect of their signs can consult the table below. It makes a difference especially for those who were born in January or Febuary which is often the end of the previous year according to the lunar calendar.
Chinese Zodiac for the past century:
1900 1.31 - 1901 2.18: Metal rat 1901 2.19 - 1902 2.7: Metal ox 1902 2.8 - 1903 1.28: Water tiger 1903 1.29 - 1904 2.15: Water rabbit 1904 2.16 - 1905 2.3: Wood dragon 1905 2.4 - 1906 1.24: Wood snake 1906 1.25 - 1907 2.12: Fire horse 1907 2.13 - 1908 2.1: Fire goat 1908 2.2 - 1909 1.21: Earth monkey 1909 1.22 - 19010 2.9: Earth rooster 1910 2.10 - 1911 1.29: Metal dog 1911 1.30 - 1912 2.17: Metal pig 1912 2.18 - 1913 2.5: Water rat 1913 2.6 - 1914 1.25: Water ox 1914 1.26 - 1915 2.13: Wood tiger 1915 2.14 - 1916 2.2: Wood rabbit 1916 2.3 - 1917 1.22: Fire dragon 1917 1.23 - 1918 2.10: Fire snake 1918 2.11 - 1919 1.31: Earth horse 1919 2.1 - 1920 2.19: Earth goat 1920 2.20 - 1921 2.7: Metal monkey 1921 2.8 - 1922 1.27: Metal rooster 1922 1.28 - 1923 2.15: Water dog 1923 2.16 - 1924 2.4: Water pig 1924 2.5 - 1925 1.24: Wood rat 1925 1.25 - 1926 2.12: Wood ox 1926 2.13 - 1927 2.1: Fire tiger 1927 2.2 - 1928 1.22: Fire rabbit 1928 1.23 - 1929 2.9: Earth dragon 1929 2.10 - 1930 1.29: Earth snake 1930 1.30 - 1931 2.16: Metal horse 1931 2.17 - 1932 2.5: Metal goat 1932 2.6 - 1933 1.25: Water monkey 1933 1.26 - 1934 2.13: Water rooster 1934 2.14 - 1935 2.3: Wood dog 1935 2.4 - 1936 1.23: Wood pig 1936 1.24 - 1937 2.10: Fire rat 1937 2.11 - 1938 1.30: Fire ox 1938 1.31 - 1939 2.18: Earth tiger 1939 2.19 - 1940 2.7: Earth rabbit 1940 2.8 - 1941 1.26: Metal dragon 1941 1.27 - 1942 2.14: Metal snake 1942 2.15 - 1943 2.4: Water horse 1943 2.5 - 1944 1.24: Water goat 1944 1.25 - 1945 2.12: Wood monkey 1945 2.13 - 1946 2.1: Wood rooster 1946 2.2 - 1947 1.21: Fire dog 1947 1.22 - 1948 2.9: Fire pig 1948 2.10 - 1949 1.28: Earth rat 1949 1.29 - 1950 2.16: Earth ox 1950 2.17 - 1951 2.5: Metal tiger 1951 2.6 - 1952 1.26: Metal rabbit 1952 1.27 - 1953 2.13: Water dragon 1953 2.14 - 1954 2.2: Water snake 1954 2.3 - 1955 1.23: Wood horse 1955 1.24 - 1956 2.11: Wood goat 1956 2.12 - 1957 1.30: Fire monkey 1957 1.31 - 1958 2.17: Fire rooster 1958 2.18 - 1959 2.7: Earth dog 1959 2.8 - 1960 1.27: Earth pig 1960 1.28 - 1961 2.14: Metal rat 1961 2.15 - 1962 2.4: Metal ox 1962 2.5 - 1963 1.24: Water tiger 1963 1.25 - 1964 2.12: Water rabbit 1964 2.13 - 1965 2.1: Wood dragon 1965 2.2 1966 1.20: Wood snake 1966 1.21 - 1967 2.8: Fire horse 1967 2.9 - 1968 1.29: Fire goat 1968 1.30 - 1969 2.16: Earth monkey 1969 2.17 - 1970 2.5: Earth rooster 1970 2.6 - 1971 1.26: Metal dog 1971 1.27 - 1972 2.14: Metal pig 1972 2.15 - 1973 2.2: Water rat 1973 2.3 - 1974 1.22: Water ox 1974 1.23 - 1975 2.10: Wood tiger 1975 2.11 - 1976 1.30: Wood rabbit 1976 1.31 - 1977 2.17: Fire dragon 1977 2.18 1978 2.6: Fire snake 1978 2.7 - 1979 1.27: Earth horse 1979 1.28 - 1980 2.15: Earth goat 1980 2.16 - 1981 2.4: Metal monkey 1981 2.5 - 1982 1.24: Metal rooster 1982 1.25 - 1983 2.12: Water dog 1983 2.13 1984 2.1: Water pig 1984 2.2 - 1985 2.19: Wood rat 1985 2.20 - 1986 2.8: Wood ox 1986 2.9 - 1987 1.28: Fire tiger 1987 1.29 - 1988 2.16: Fire rabbit 1988 2.17 1989 2.5: Earth dragon 1989 2.6 - 1990 1.26: Earth snake 1990 1.27 - 1991 2.14: Metal horse 1991 2.15 - 1992 2.3: Metal goat 1992 2.4 - 1993 1.22: Water monkey 1993 1.23 - 1994 2.9: Water rooster 1994 2.10 - 1995 1.30: Wood dog 1995 1.31 - 1996 2.18: Wood pig 1996 2.19 - 1997 2.6: Fire rat 1997 2.7 - 1998 1.27: Fire ox 1998 1.28 - 1999 2.15: Earth tiger 1999 2.16 - 2000 2.4: Earth rabbit 2000 2.5 - 2001 1.23: Metal dragon 2001 1.24 - 2002 2.??: Metal snake