[A] p-value is the probability of our observing the sample data we have observed when, in fact, the two population averages are identical. - Anthony Davies, Understanding Statistics (2017)

An Intuitive Explanation

Imagine you are comparing two observations (a la some difference of means test): the mean and standard deviation of employment rate in California and New York, USA.

You can think of the p-value as the probability that the apparent difference in employment rate between California and New York is due to random chance.

Think about this:

The higher that probability, the more likely any disparity can be (effectively) written off.
The lower that probability, the more significant the difference is and more you should consider investigating further.

Closely related to this idea is statistical significance, where a p-value of 0.05 is typical and 0.01 is used in more stringent trial.

Mathematical explanation

link into a statistics learning path

equation from mean-value test

P-value