This lemma gives an optimum decision procedure for hypothesis testing. Suppose you have 2 hypotheses, H0 and H1. You wish to know which, in fact, holds, so you perform an experiment and observe the results O (for our purposes, O can be the result of a series of experiments). The Nayman-Pearson lemma tells you that you should check if this holds (the parameter t will be explained later):
P(H1 | O) / P(H0 | O) > t
If it holds, you should decide the hypothesis H1 holds; otherwise H0 (sometimes called the null hypothesis) holds. Of course, larger values of t make it harder for you to decide the H1 holds. The Nayman-Pearson lemma says that for every (positive) value of t, this procedure is optimal in the following sense:
Thus, we see that the value t controls the ratio of false negatives to false positives. Different applications will require different ratios; we choose it to give the desired ratio.