It means, that the correlation (the relationships between x and y) is not stationary. It cahnges in time/spatial.

Zoli

It means that you have decided to act as if x and y have a different correlation coefficient in the different age groups.

"Significance" is something you assign. Results are not per se significant. Results are results. "Significance" is a judgment. It is usually oriented at the p-value of a null-hypothesis test, and a result is judged "significant" when the p-value is "low". Again, how low is low enough to judge the result "significant" is eventually a personal decision.

The p-value tells you how likely you would expect such data (ore more "extreme" data) under the specified null hypothesis. The null-hypothesis in this case is presumably: the correlation coefficient (r) between x and y is the same in both age groups.

One can invent a "test statistic" to sensitively express deviations from this null hypothesis, for instance the ratio of the two r-values. One can then further elaborate what one can expect how this statistic will be distributed given it is obtained from noisy data - given the null-hypothesis is true. In the absence of "noise" the value would be 1, exactly. But in the presence of noise the observed values can be different. Most likely the values will be close to 1, but they may also be larger or smaller. Given a certain amount of data and noise, one can calculate the probabilities of getting ratios that are below, say, 0.2 or greater than 5 etc. Now you have your data from which you can estimate the "noise" and you get an observed ratio (test statistic). You can then calculate how likely you expect a test satistic under the null-hypothesis that is at least as different from 1 than the one you observed. This is the p-value. A low p-value says: You would not expect such data if the null hypothesis was true.

Fisher's transformation is a transformation of r to get a transformed variable with an approximate normal distribution. Instead of investigating the ratio of r-values Fisher's test investigates the difference between these transformed values. This has the advantage that the sampling distribution of the test statistic (i.e. the difference of these transformed values) under the null hypothesis is known and there are standard tools available to get the p-values.

Thanks for the responses, big help in interpreting the results..