Best critical region and Neyman-Pearson lemma?

I'm reading an online course and I'm really confused about the Neyman-Pearson lemma. It states From my understanding, does it mean that critical region could be any region like (a,b) and (c,+00)? And I'm very confused with most powerful critical region according to its definition what does K mean? Can K be any constant number?

• It means a constant number, which you need to determine so that the test has the required significance level. – JohnK Jan 28 '16 at 23:23
• I posted an illustrated account of this theory at stats.stackexchange.com/a/130772. – whuber Jan 29 '16 at 0:33

Both of the regions you shaded correspond to $\alpha=0.05$, assuming that your plot represents the null distribution.
The best critical region would be the one of these that would be most likely under the alternative hypothesis. If the alternative hypothesis was $H_a:\mu = C$ for any $C>0$, then the tail critical region would be the best critical region because it would be the critical region with the largest probability under the alternative.