Broadly, there are two kinds of error: false positives and false negatives. A false positive arises because reality conspired to produce what you consider to be an anomalous result. False negatives arise because your detector wasn't sensitive enough to notice the anomalous result.

The standard statistical practice is to fix your risk of false positives through the preemptive choice of a p-value threshold and then to attempt to minimize your risk of false negatives through increases in the "power" of your experiment. But, honestly, these two are in tension with one another. There are lots of possible choices.

It's important to know the difference, though. It changes how you interpret results. A positive result might be doubted because you, as a reader, would have preferred more protection against false positives, and, thusly, a smaller p-value threshold. A negative result might be doubted because you believe that the experimenter under-powered their experiment. You might also, in a repeating experimental process, wish to obtain greater power by sacrificing false-positive protection. This is as easy as changing the p-value threshold.

A last perspective is that in the typical scientific practice, you basically control false-positives through pushing the p-value down and making the experiment more challenging (reducing power). Then, you re-establish power through more expensive experiments (use better tools, run the experiment for longer, improve controls, improve data efficiency). Often the "typical" p-value is set by the community in relation to what level of false-positive the overall community is willing to tolerate in publication. That's why psychologists are happy with a 5% threshold whereas physicists might seek a threshold of 10^-3 or smaller. The physicists are upholding themselves to a much more expensive experimental standard and would vastly prefer false negatives.