Chinese Zodiac of the year

Pyth - 65 bytes

@c." y=CÅQ¯@îR
°©ÀáëªfÞ­#[õNqÀN8Ô'¼ô,n7˶~fB").d3

Try it online here.

@                     Modular indexing
 c           )        Split string, use default of spaces
  ."ajs&^asd"         Packed string to compress space separated zodicas
 .d3                  Current year

Pyth, 54 bytes

@rR3c."ayÖÚ©*ÐPk<`eüµÜxþ!¬Ì ¾½hÑðÙBwp"\c.d3

Try it online: Demonstration

You can replace the last 3 chars .d3 with any year number to test the other Zodiac animals.

Explanation:

     ."ayö..."        packed string, decrypts to "monkeycroostercdogcpig..."
    c         \c      split this string at "c"s
 rR3                  make the first letter of each animal uppercase
                .d3   current year
@                     take the correct animal of the list and print

JavaScript ES6, 106 105 104 103 bytes

Saved 1 byte thanks to Maltysen, 1 byte to insertusernamehere, and 1 byte to edc65!

x=>"RatOxTigerRabbitDragonSnakeHorseGoatMonkeyRoosterDogPig".match(/.[a-z]+/g)[new Date().getYear()%12]

Pretty self explanatory.