An odds ratio gives you a way of measuring how strongly and event $A$ is associated to an event $B$.

This is best understood using an example. Suppose you've created a home made face cream but some of the friends you've given it to said they got spots after using it. The odds ratio gives you a way of measuring how strongly people getting spots is associated to using the face cream, using available data.

Let's suppose that you gave the face cream to 10 people, 6 of whom developed spots within a week. The odds of getting spots, having used the face cream, is the ratio between the number of people who got spots and the number of people who didn't, so that's $6/4=3/2$.

Since people might develop spots spontaneously for other reasons too, you ask the friends you haven't given the face cream to if they also developed spots within a week. Suppose that there are 20 people in total you haven't given the cream to, and that 5 of them did develop spots. The odds of developing spots, having not used the cream, is therefore $5/15=1/3$.

The odds ratio compares the odds of spots with the cream to the odds of spots without the cream. It does so by taking the ratio of the former to the latter: $$\text{odds ratio}=\frac{3/2}{1/3}=\frac{3\times 3}{2}=\frac{9}{2}=4.5.$$ Since this number is greater than $1$ you know that the odds of getting spots having used the cream is bigger than the odds of spots not having used the cream — in fact, it's more than four times as big. This indicates an association of using the cream and getting spots. Were the ratio less than $1$, then this would indicated an association between not using the cream and getting spots. And were it exactly equal to $1$, then this might indicate that whether or not you get spots has nothing to do with the cream.

Note that an odds ratio does not deliver conclusive proof that one event causes another, for example that the cream causes spots. Even though in our example the odds ratio indicates an association between the cream and spots, this association could simply be down to chance. Or it could be down to other factors. For example, you may have chosen to give the face cream to people you know have problem skin, in which case them having a higher odds of developing spots is not surprising. It's important not to confuse correlation with causation. To get conclusive answers as to what your cream does to skin you need to do more research.

Let's now look at the general formula for working out an odds ratio to see how strongly event $A$ is associated to event $B$. Suppose that in $a$ outcomes of a process event $A$ occurred and event $B$ also occurred, and that in $b$ outcomes event $A$ didn't occur and event $B$ occurred. The odds of $A$ occurring alongside $B$ is therefore $a/b$.

Also suppose that in $c$ outcomes of the process event $A$ occurred and event $B$ didn't occur, and that in $d$ outcomes event $A$ didn't occur and event $B$ didn't occur. The odds of $A$ occurring in the absence $B$ is therefore $c/d$.

The odds ratio here is $$\mbox{odds ratio} = \frac{a/b}{c/d} = \frac{ad}{bc}.

Note that the word "event" here is used in a very general sense. For example, if you had a group of cats and dogs and wanted to see whether catching mice is something associated to cats more than dogs, then "being a cat" and "being a dog" would be classed as events.

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