When analyzing a contingency table, the p value indicates a significant difference (P<0.05), but the confidence interval for the Odds Ratio or Relative Risk does not. Why?

P values and confidence intervals are intertwined. If the P value is less than 0.05, then the 95% confidence interval cannot contain the value that defines the null hypothesis. (You can make a similar rule for P values < 0.01 and 99% confidence intervals, etc.)

This rule is not always upheld with results from contingency tables.

The P value computed from Fisher's test is exactly correct. However, the confidence intervals for the Odds ratio and Relative Risk are computed by methods that are only approximately correct. Therefore it is possible that the confidence interval does not agree with the P value. For example, it is possible for results to show P<0.05 with a 95% CI of the relative risk that includes 1.0. (A relative risk of 1.0 means no risk, so defines the null hypothesis). Similarly, you can find P>0.05 with a 95% CI that does not include 1.0.

These apparent contradictions happens rarely, and most often when one of the values you enter equals zero.