A measure of how surprising your result would be if nothing were really going on. It is routinely misread as the probability that your finding is true, which it is not. A significant result can be tiny and useless; a non-significant one can mean you simply did not gather enough data.
Technical view
The probability of observing data at least as extreme as yours under the null hypothesis. It says nothing about effect size or about the probability of the hypothesis. Multiple comparisons inflate false positives, requiring correction. In evaluation work the practical issues are running many variants until one passes, and treating a threshold as a decision rule rather than reporting effect size with an interval.
Related
Part of AI Terminology Index, section: Maths and Statistics.