Relationship between statistical power and beta.

Power

Even if the treatment affects the outcome, you might not obtain a statistically significant difference in your experiment. Just by chance, your data may yield a P value greater than alpha.

Let's assume we are comparing two means with a t test. Assume that the two means truly differ by a particular amount, and that you perform many experiments with the same sample size. Each experiment will have different values (by chance) so a t test will yield different results. In some experiments, the P value will be less than alpha (usually set to 0.05), so you call the results statistically significant. In other experiments, the P value will be greater than alpha, so you will call the difference not statistically significant.

If there really is a difference (of a specified size) between group means, you won't find a statistically significant difference in every experiment. Power is the fraction of experiments that you expect to yield a "statistically significant" P value. If your experimental design has high power, then there is a high chance that your experiment will find a "statistically significant" result if the treatment really works.

Beta

The variable beta is defined to equal 1.0 minus power (or 100% - power%). If there really is a difference between groups, then beta is the probability that an experiment like yours will yield a "not statistically significant" result.  

Don't mix up this use of beta, with the beta function used by mathematicians and mathematical statisticians.