How does the effect size inform the design (or analysis) of an NHST?

Consider this answer on how to design an NHST. I don’t quite understand what exact process one is supposed to follow to determine the minimum sample size once we have:

• A null hypothesis that is falsifiable. The choice of this null hypothesis will inform the assumed distribution of the data under the null.
• A minimum effect size that we want to be able to observe.
• A maximum false positive rate, i.e. the significant level $alpha$ against which we’ll compare a computed $p$ value.
• The maximum false negative rate = 1-$beta$, i.e. where $beta$ is called the power or sensitivity of the experiment.

If simpler, let’s assume a 1D Normal distribution for the Null.

My recollection is that one is supposed to use the CDF of the Null somehow for the test, but how? E.g. how does the chosen value for the minimum effect size inform the sample size?