What exactly is the test called for the significance of a correlation? I am not sure if it even has a name, but when we look at the p-value for example from scipy.stats.pearsonr what is the test name that is used to test for the null hypothesis?
 A: To find the statistical significance of a Pearson r correlation use Fisher's z transformation and see if the confidence intervals includes your hypothesized value. For example, the two-tailed 95% confidence interval of a Pearson r is 
$$0.5\ln\mid{\frac{1-r}{1+r}}\mid\pm1.96\frac{1}{\sqrt{n-3}}$$
A: There's more than one possible test of a Pearson correlation coefficient. 
Generically I'd just call it a test of a Pearson correlation.
The test in scipy.stats.pearsonr is predicated on the assumption of joint normality and is for testing the specific null that the population correlation is 0 against either a one- or two-tailed alternative (typically the second). 
This would result in a test based on a symmetric beta distribution for the sample correlation under $H_0$ but it can be turned into a test based on a t-distribution (and usually is, but by the look of it scipy.stats.pearsonr sticks with the direct calculation based on the symmetric beta).
You might call it a test of a null Pearson correlation under joint normality, perhaps.
As noted in the Wikipedia article at the above link, nonparametric tests of a null of 0 correlation are possible (e.g. it described a permutation test). Under some sufficiently defined circumstances, other parametric tests (besides the one for joint normality) are possible as well.
There's also tests for the case where the correlation specified under the null is not zero (again, see the Wikipedia link); these can be based on the exact distribution or one based one Fisher's z-transform, but neither of these appear to be implemented in scipy.stats.pearsonr.
