A contingency table is a table listing the frequency of different combinations of two categorical variables. For example, you could have a contingency table of sex and whether the person voted for McCain, Obama or neither. This would be a 2x3 contingency table. The odds ratio is a measure of the strength of relationship between two categorical variables. More specifically, it is the ratio of the odds of each category on one variable doing something on the other variable; in the example, you could have the ratio of the odds of men voting for McCain vs. Obama to the odds of women voting for McCain vs. Obama. The odds ratio can only compare four cells of a contingency table.
Decide which four cells you want to compare. These must be in a particular square. For example, in a table of vote (McCain, Obama, neither) versus sex (male, female) you could choose to use McCain and Obama but not other.
Multiply the number in row 1, column 1 by the number in row 2, column 2. In the example, if men are in row 1 and women in row 2, while McCain votes are in column 1 and Obama votes in column 2, this would be multiplying the number of men voting for McCain by the number of women voting for Obama.
Multiply the number in row 1, column 2 by the number in row 2, column 1. In the example, this would be the number of men voting for Obama times the number of women voting for McCain.
Divide the result in Step 2 by the result in Step 3. This is the odds ratio.