A legend explains the 12 animals chosen:

Buddha summoned all the animals to come to him before he departed from earth. Only twelve came and as a reward he named a year after each one in the order they arrived.

Note that there is more than one way to translate each animal into English: pig is also boar, snake is also serpent, sheep is also ram, rabbit is also hare.

On which year was I born?

The following is a table of years from 1924 (assuming nobody here is older than that) to 2019 (if it is now past 2019, /msg me or my next of kin, and we'll update it):

  
Rat      1924  1936  1948  1960  1972  1984  1996  2008
Ox       1925  1937  1949  1961  1973  1985  1997  2009
Tiger    1926  1938  1950  1962  1974  1986  1998  2010
Rabbit   1927  1939  1951  1963  1975  1987  1999  2011
Dragon   1928  1940  1952  1964  1976  1988  2000  2012
Snake    1929  1941  1953  1965  1977  1989  2001  2013
Horse    1930  1942  1954  1966  1978  1990  2002  2014
Sheep    1931  1943  1955  1967  1979  1991  2003  2015
Monkey   1932  1944  1956  1968  1980  1992  2004  2016
Rooster  1933  1945  1957  1969  1981  1993  2005  2017
Dog      1934  1946  1958  1970  1982  1994  2006  2018
Boar     1935  1947  1959  1971  1983  1995  2007  2019
What does each animal mean?

Here is a short outline of the main characteristics of each animal: