First weighing: four against four
Second weighing: two against two
Third weighing: one against one

Let’s name the balls 1-12.

First we weigh {1,2,3,4} on the left and {5,6,7,8} on the right. There are three scenarios which can arise from this.

If they balance, then we know 9, 10, 11 or 12 is odd. Weigh {8, 9} and {10, 11} (Note: 8 is not odd)

If they balance, we know 12 is the odd one. Just weigh it with any other ball and figure out if it is lighter or heavier.

If {8, 9} is heavier, then either 9 is heavy or 10 is light or 11 is light. Weigh {10} and {11}. If they balance, 9 is odd (heavier). If they don’t balance then whichever one is lighter is odd (lighter).

If {8, 9} is lighter, then either 9 is light or 10 is heavy or 11 is heavy. Weigh {10} and {11}. If they balance, 9 is odd (lighter). If they don’t balance then whichever one is heavier is odd (heavier).

If {1,2,3,4} is heavier, we know either one of {1,2,3,4} heavier or one of {5,6,7,8} is lighter but it is guarantees that {9,10,11,12} are not odd. Weigh {1,2,5} and {3,6,9} (Note: 9 is not odd).

If they balance, then either 4 is heavy or 7 is light or 8 is light. Following the last step from the previous case, we weigh {7} and {8}. If they balance, 4 is odd (heavier). If they don’t balance then whichever one is lighter is odd (lighter).

If {1,2,5} is heavier, then either 1 is heavy or 2 is heavy or 6 is light. Weigh {1} and {2}. If they balance, 6 is odd (lighter). If they don’t balance then whichever one is heavier is odd (heavier).

If {3,6,9} is heavier, then either 3 is heavy or 5 is light. Weigh {5} and {9}. They won’t balance. If {5} is lighter, 5 is odd (lighter). If they balance, 3 is odd (heavier).

If {5,6,7,8} is heavier, it is the same situation as if {1,2,3,4} was heavier. Just perform the same steps using 5,6,7 and 8. Weigh {5,6,1} and {7,2,9} (Note: 9 is not odd).

If they balance, then either 8 is heavy or 3 is light or 4 is light. We weigh {3} and {4}. If they balance, 8 is odd (heavier). If they don’t balance then whichever one is lighter is odd (lighter).

If {5,6,1} is heavier, then either 5 is heavy or 6 is heavy or 2 is light. Weigh {5} and {6}. If they balance, 2 is odd (lighter). If they don’t balance then whichever one is heavier is odd (heavier).

If {7,2,9} is heavier, then either 7 is heavy or 1 is light. Weigh {1} and {9}. If they balance, 7 is odd (heavier). If they don’t balance then 1 is odd (lighter).

Note: There are other possible solutions to this problem as well.