What exactly does a "significant correlation" mean? I performed the correlation between several variables. R makes it possible to separate significant from non-significant correlations.
I noticed that low corrections (corr=0.4) can be considered significant (p.value<0.05), while non-significant corrections (p.value>0.05) can assume relatively high values (corr=0.7 ).
In this sense, I would like to know what exactly it means to say that one correlation is more significant than another? what arguments could I use to explain the situation above?

If you write down the formula for calculating the test statistic, you’ll see that sample size is a factor.
If you have a small correlation value but a large sample size, you can wind up with a small p-value, indicating that there is strong evidence that the correlation is not zero. Even if the true correlation is trivially different from zero, such as $0.01$, that number is not zero.
If you have a small sample size, then you might need a very gigantic correlation value to get the hypothesis test to return a small p-value.
Like all hypothesis testing, sample size plays a huge role in testing correlation.
For further reading, you might want to look into the difference between practical and statistical significance, and you might be interested in equivalence testing.
