Short Definition
A p-value measures how surprising observed data would be if a specified null hypothesis were true.
Intuition
A small p-value says the data would be unusual under the null model, not that your hypothesis is automatically true.
Technical Definition
The p-value is the probability, under the null hypothesis, of observing a test statistic at least as extreme as the one observed.
Example
An experiment may report p = 0.03 for a difference in conversion rates under a no-difference null hypothesis.
Common Misunderstandings
A p-value is not the probability the null hypothesis is true.
Statistical significance is not practical significance.