You should consider using lookup tables when possible. That increment loop can also be replaced by something more direct.
If you can't use a table, a switch statement is heavily preferred over a series of if's like this (and will likely be more efficient).
Code:
#include <iostream>
using namespace std;
int main()
{
int y;
cout<<"Enter your year of birth: ";
cin>>y;
if(y < 0)
y = 12 - (-y % 12);
if (y == 14) {cout<<"Really? Me, too!";}
char *signs[] = { "Monkey", "Chicken", "Dog", "Pig", "Rat", "Ox", "Tiger",
"Rabbit", "Dragon", "Snake", "Horse", "Sheep" };
cout << "You were born in the year of the " << signs[y % 12] << ".";
cout<<"\n\n";
system("PAUSE");
}If you can't use a table, a switch statement is heavily preferred over a series of if's like this (and will likely be more efficient).