Pascal, 181 B

This complete program require a processor supporting features of Extended Pascal as defined by ISO standard 10206, in particular the time‑related routines and data type.

program z(output);var t:timeStamp;begin
getTimeStamp(t);write(subStr('Monkey RoosterDog    Pig    Rat    Ox     Tiger  Rabbit Dragon Snake  Horse  Goat   ',t.year mod 12*7+1,7))end.

Long:

program chineseZodiacOfTheYear(output);
    const
        animal = 'Monkey RoosterDog    Pig    Rat    Ox     Tiger  Rabbit Dragon Snake  Horse  Goat   ';
    var
        now: timeStamp;
    begin
        getTimeStamp(now);
        writeLn(output, trim(subStr(animal, now.year mod 12 * 7 + 1, 7)))
    end.

See also: RosettaCode task

Vyxal, 49 bytes

`⌊ẇ ₃¼ Roos≠∴ •∞ Ḣ⟩ ₀⌊ Ox ṗȦ ₈Ė ėɾ Ċṅ ½√`⌈kD⌊12%i

Try it Online!

`...`         # Compressed  string of names
     ⌈        # Split on spaces
      kD⌊     # Current year
         12%  # Modulo 12
            i # Index into list of names

J, 78 bytes

(>;:'Monkey Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Horse Goat'){~12|

Usage:

    (>;:'Monkey Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Horse Goat'){~12| 2015
 Goat

Explanation:

;:      separates the string into boxed array according to the spaces
>       opens the boxed array so that it becomes a normal array
x {~ y  is choose the y-th element from x.
        the ~ reverses the argument, so x { y is choose the x-th element from y.
12|     is the residue when the year is divided by 12

Python 3, 126 116 112 bytes

Edit: I'm keeping the other answer because it's cool, but this is shorter.

from time import*
"Monkey Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Goat".split()[localtime().tm_year%12]

If trailing whitespace is ok, then it's 126 bytes. If not, it's 134 bytes after adding .strip() to the end of the code.

from datetime import*
"MRDPROTRDSHGoooiaxiarnoonoggt gbaaraks    ebgkstet    rioee ye     tn    r"[datetime.now().year%12::12]

C# 159

using System;class P{static void Main(){Console.Write("Monkey;Rooster;Dog;Pig;Rat;Ox;Tiger;Rabbit;Dragon;Snake;Horse;Goat".Split(';')[DateTime.Now.Year%12]);}}

Readable / ungolfed version

using System;
class Program
{
    static void Main(string[] args)
    {
        Console.Write("Monkey;Rooster;Dog;Pig;Rat;Ox;Tiger;Rabbit;Dragon;Snake;Horse;Goat".Split(';')[DateTime.Now.Year % 12]);
    }
}

PowerShell, 92 bytes

(-split"Monkey Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Horse Goat")[(date).Year%12]

Similar in spirit to other answers, added here only for sake of cataloging. This takes the string "Monkey ... Goat" and -splits it on spaces to create an array, which is cheaper than specifying the elements individually with ("Monkey","Rooster",...). We index into that array with the current year mod 12, which outputs the result. Nothing too fancy.

Python 129 126 bytes

EDIT: I just learnt s.split(' ')=s.split()

Although this isn't the shortest method, here is an alternative answer to @Sherlock9's one:

from datetime import*
print 'Monkey Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Horse Goat'.split()[date.today().year%12]

Try it here

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.

Japt, 72 65 bytes

`Mkey RooÐP Dog Pig R Ox Tig Rabb Dg Snake HÆ  Go`¸gÐ i %C

Try it online!

CJam, 69 bytes

"SKiN<lEz=2)aFBY8H[$h-0|4=K`T2Cp%-a)o6"132b26b'`f+'`/et0==(eu\

Contains a bunch of unprintables. Try it here.

PHP >= 5.6, 84 bytes

Very straight forward:

<?=[Monkey,Rooster,Dog,Pig,Rat,Ox,Tiger,Rabbit,Dragon,Snake,Horse,Goat][date(Y)%12];

R, 119 bytes

scan(t="Monkey Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Horse Goat",w="")[(format(Sys.Date(),"%Y"):1)[1]%%12+1]

format(Sys.Date(),"%Y") returns the year as a character string, (format(Sys.Date(),"%Y"):1)[1] is the shortest way i could think of to coerce it into an integer so that we can apply %% to it (the sequence generator : indeed coerces automatically to integer). +1 because R indices are 1-based and not 0-based.

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

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

Milky Way 1.6.1, 101 96 bytes

M" "\=A12n"Goat Horse Snake Dragon Rabbit Tiger Ox Rat Pig Dog Rooster Monkey"" "\;>=<&{~1-;^};!

Saved 5 bytes thanks to a reminder by Luis Mendo.

Explanation

The code modulos the system year by 12 and outputs the proper animal.

Usage

./mw <path-to-code>

MATL, 85 bytes

'Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Horse Goat Monkey'YbZ'10H$XOU12X\)

Uses release 6.0.0 of the language/compiler. Works in Octave.

Example

>> matl
 > 'Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Horse Goat Monkey'YbZ'10H$XOU12X\)
 >
Monkey

Explanation

Note that the initial string is cyclically displaced by 3 to avoid a subtraction.

'Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Horse Goat Monkey'
Yb                    % split at spaces
Z'                    % number representing current date and time
10H$XO                % get year as a string
U                     % convert to number
12X\                  % modulo 12, with 0 converted to 12
)                     % index (and implicitly print)

CJam, 82 80 75 bytes

This is was a direct port of my Milky Way answer.

"Monkey Rooster Dog Pig Rat Ox Tiger Rabbit Dragon Snake Horse Goat"S/et0==

Saved 7 bytes thanks to NinjaBearMonkey.