How to interpret Fisher Test? How do I interpret the odds ratio for this fisher test output in R?
...below is my matrix that I input into fisher.test in R:

...and below this is the output of the test in R:

 A: That p-value is for the null hypothesis of independence between the two categorical variables.  We reject the null of independence here.
For the odds ratio, if the confidence interval contains one, we fail to reject the null hypothesis of independence.
As you can see, we reject the null of independence since the confidence interval doesn't contain 1.
Update: The section below is a general description to help you understand what the odds ratio is and why a value close to 1 means we don't have evidence against independence (that is, loosely speaking, why a value close to 1 should be associated in your mind with independence)
Odds ratio $\theta$ is defined as $\theta = \frac{n_{11}/n_{12}}{n_{21}/n_{22}}=\frac{n_{11}n_{22}}{n_{12}n_{21}}$.  
If the ratio of counts in row 1 is very similar to the ratio of counts in row 2, the odds ratio will be close to 1.  In the case that $\theta$ is close to one, the "row you are in" doesn't seem to affect the column count very much.  Hence, there is a (suggested) lack of association, which is to say (a suggestion of) independence.
A: The 95%-CI for the odds ratio spans from 0.000 to 0.6844, thus an odds ratio of 1.0 is far off and you can reject the null hypothesis, that the values in the contingency table are independent. p = 7.9e-12 means $7.9 \times 10^{-12}$, so very close to zero. Conventionally, the test result is considered 'significant' if p < 0.05. Here p is smaller than that by many magnitudes.  
