Significance tests for correlations
There are tests of statistical significance that can be applied to individual correlations, which indicate the probability of obtaining a correlation as large or larger than the the sample correlation assuming the null hypothesis is true.
The key point is that what constitutes a statistically significant correlation coefficient depends on:
- Sample size: bigger sample sizes will lead to smaller thresholds
- alpha: often set to .05, smaller alphas will lead to higher thresholds for statistical significance
- one-tailed / two-tailed test: I'm guessing that you would be using two-tailed so this probably doesn't matter
- type of correlation coefficient: I'm guessing you are using Pearson's
- distributional assumptions of x and y
In common circumstances, where alpha is .05, using two-tailed test, with Pearson's correlation, and where normality is at least an adequate approximation, the main factor influencing the cut-off is sample size.
Threshold of importance
Another way of interpreting your question is to consider that you are interested not in whether a correlation is statistically significant, but rather whether it is practically important.
Some researchers have offered rules of thumb for interpreting the meaning of correlation coefficients, but these rules of thumb are domain specific.
Multiple significance testing
However, because you are interested in flagging significant correlations in a matrix, this changes the inferential context.
You have $k(k-1)/2$ correlations where $k$ is the number of variables (i.e., $14(13)/2=91$.
If the null hypothesis were true for all correlations in the matrix, then the more significance tests you run, then the more likely you are of making a Type I error. E.g., in your case you would on average make $91 * .05 = 4.55$ Type I errors if the null hypothesis were true for all correlations.
As @user603 has pointed out, these issues were well discussed in this earlier question.
In general, I find it useful when interpreting a correlation matrix to focus on higher level structure. This can be done in an informal way by looking at general patterns in the correlation matrix. This can be done more formally by using techniques like PCA and factor analysis. Such approaches avoid many of the issues associated with multiple significance testing.