In frequentist hypothesis testing, the $p$-value is the probability of a result as extreme (or more) than the observed result, under the assumption that the null hypothesis is true.
In frequentist hypothesis testing, the $p$-value is the probability of a result as extreme (or more) than the observed result, under the assumption that the null hypothesis is true. (Extremity is defined with respect to the likelihood ratio of the alternative vs. the null hypothesis. Hence, extremity depends on the alternative.) When the $p$-value is small, the observed data would be unlikely to occur if the null were true. This fact is then typically used to argue that the null hypothesis is false. That is, a low p-value is typically interpreted as evidence suggesting the null hypothesis is false. The most common cutoff for a "small" $p$-value in research is $.05$.
