In plain English what is the difference between a most powerful test and a uniformly most powerful test? I'm having trouble understanding the two concepts of a powerful test and a uniformly powerful test. I'm reading about these tests in context of the Neyman Pearson Lemma and it seems like they're virtually the same thing? 
 A: "Uniformly" means regardless of the values of the unobservable parameters. One test may be the most powerful one for a particular value of an unobservable parameter while a different test is the most powerful one for a different value of the parameter. A uniformly more powerful test remains the most powerful one regardless of the value of the parameters.
A: According to Mood [1]


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*Most powerful test (MP): a test $\delta$ of $H_0: \theta = \theta_0$ vs $H_1: \theta = \theta_1$ of size $\alpha$ is MP if it has the greatest power $\pi(\theta_1 \mid \delta)$ among all tests of size $\alpha$ or less, where $\pi$ is the power function. In words, $\delta$ has the greatest capacity of detecting $H_1$ among tests of size at most $\alpha$ and these specific hypothesis.

*Uniformly most powerful test (UMP): a test $\delta$ of $H_0: \theta \in \Theta_0$ vs $H_1: \theta \in \Theta - \Theta_0$ size $\alpha$ is UMP if it has the greatest power $\pi_{\theta \in \Theta_1}(\theta \mid \delta)$ among all tests of size $\alpha$ or less. "Uniformly" refers to all values of $\theta$.


Notice the difference in the two staments with respect to the hypothesis and power. A non-UMP test can be most powerful just for a specific value of $\theta$. A UMP test is is the "most powerful" test for each value of $\theta$ in $H_1$.
[1] Mood, Alexander McFarlane. "Introduction to the Theory of Statistics." (1950).
