The p-value of a hypothesis test is the smallest value of alpha (significance level) that would lead to rejection of the null hypothesis. It is the observed level of significance.
A small p-value provides evidence in support of the alternative hypothesis. In practice, the analyst sets the desired level of significance to, say, 5%. If the observed significance level (the p-value) is less than the desired level of significance then the null hypothesis can be rejected.
For instance, following the example under concept "One-side tests" (on real estate) the p-value is the probability
p-value = P(z > 1.932) = 0.0316
For a given significance level of 5%, the observed p-value of 0.0316 indicates that the null hypothesis can be rejected. It can even be said that there is a 3.16% probability of making a mistake by rejecting the null hypothesis (less than the tolerated level of 5%).