Naive question on p value

I just found out that correlation coefficient is the square root of
the r-squared. I understand the latter but never quite know what the
former was. Another statistics term "p value" also is pretty murky to
me - I'm guessing for the same reason correlation coefficient was
murky, i.e. lots of hoopla and wordy explanation of it. I'm wondering
if there a technical definition of it (or <gasp> a formula I can look
at)?

I know it's the probability of obtaining the observed correlation
under the null hypothesis (or something like that ;) but I'd like to
know the actual equation. Can anyone advise? Thanks.
.