Power, Confidence, Error, Significance

Reject null hypothesis when true (false positive) = alpha or Type 1 error
Accept null hypothesis when false (or false negative) = beta or Type 2 error
Reject null hypothesis when false: POWER = (1-beta)
Accept null hypothesis when true : CONFIDENCE (= 1-alpha)
At high power, beta is small => alpha is large => likely that p-value will be < alpha (significance level). Most effects tend to be deemed significant.
At low power, beta is large => alpha is small => likely that p-value will be > alpha (significance level). Most effects tend to be deemed insignificant.