The p value is the probability of making a certain observation - or more extreme - due to chance, given that the observation is drawn from the original distribution.
Normally (haha) you would expect observations to be pretty close to the distribution's arithmetic mean. If an observation is far from the mean it can be because of two things: a) the observation occurred due to chance - this has to be possible; otherwise the distribution's probability density function would be 0 away from the mean. Or b) the observation is different because it is generated by a different distribution with a mean that differs from the original one. So a small p value means that an observation that far from the original distribution's mean is super unlikely, if it was generated by that distribution. Totally possible but really unlikely (i.e. with probability p).
Now there is this convention in one branch of the statistics community to consider p values under a certain threshold to be evidence that the observation is from a distribution different from the original one. That threshold is called "alpha" and is set prior to an experiment. So if alpha e.g. 0.05 and p is below that, we think that this is evidence that the observation is from a different distribution.
That's the story for p values for means. When it comes to correlations, the strength of the correlation (which is the observation in this case) is given by a different variable - e.g. r. Having a high r means the correlation is strong. But even high values of r can have different values of p - high ones and low ones. With a low p value we think that it is unlikely to get that kind of r due to chance. A high p value can be sign for having a great amount of noise in the data so that the correlation itself is high but the probability that this is due to chance is also high. You can also have very low values of r with high certainty (low values of p). So the strength of correlations is not measured by p.