The power of study does depend on the study design. The links below may help you

http://en.wikipedia.org/wiki/Statistical_power

http://www.indiana.edu/~statmath/stat/all/power/power.pdf

The statistical power of a study is the power, or ability, of a study to detect a difference if a difference really exists.   It depends on two things:  the sample size (number of subjects), and the effect size (e.g. the difference in outcomes between two groups).   For common studies involving comparing two groups, for example blood pressure levels between smokers and non-smokers, the T-test is usually used and the power of the study is relatively easy to compute if you know the sample size and the (hypothesized) difference in blood pressure between the two groups.  

Many small studies of this type are under-powered to detect a true difference because they do not have enough subjects, and researchers end up with a large "insignificant" p-value, but the lack of significance is really a sample size issue and not an effect size issue. 

There is the free software package G*Power that will help you compute power.  It also lets you determine the necessary effect size, or the sample size, for a given power.   This is useful for planning sample size for a study as well as post-hoc analysis of studies to see if they had enough power. (however the manual is not great so it is tricky to learn).     

Generally, a power of .80 (80 percent) or higher is considered good for a study.  This means there is an 80 percent chance of detecting a difference as statistically significant, if in fact a true difference exists.   (i.e., a difference between group A and group B, smokers and non-smokers, etc. )  

Power is computed differently of course for different kinds of analysis.  For logistic regression, there is no way to compute sample size or power when you are using continuous variables.  

The higher the power of a study is, the more subjects there are and/or the larger the effect size will be (or the smaller the p-value too).   With a very large number of subjects, a study may have good power to detect even a very small effect size---so small that it is not clinically meaningful, such as a 3-point difference in blood pressure between group A and group B.    To detect a small difference between groups, you need a higher number of subjects to have adequate power. To detect a big difference between groups, a smaller sample size will be sufficient to give adequate power.  

And of course, when planning or analyzing a study, you must keep in mind the types of potential bias, the limitations, types of validity, etc.   as well as statistical power and the factors that influence it: effect size, sample size and p-value.     

Dear Jerry, my thoughts; yet i was to sleepy yesterday night to post it.

Power is the ability to tell a difference between two groups when a difference really exists.

In small studies with limited power, authors frequently make the mistake of saying treatment A is just as good as treatment B. They are saying there is no difference, or A=B.

If the study were adequatelly powered, you might find that A and B really are different; one might be better than the other. You just do not have enough sample size to find that difference. (Or, you can find a large difference, like 20 mm Hg for blood pressure between A and B, but not a small difference like 5 mm Hg).

The power of a binary hypothesis test is denoted by "(1−β)" and is the probability of a "true positive". It is the probability that the test correctly rejects the null hypothesis when a specific alternative hypothesis is true. It is the probability of avoiding a "false negative", otherwise known as a type II error. The statistical power ranges from 0 to 1, and as statistical power increases, the size of "β" (the probability of making a type II error by wrongly failing to reject the null hypothesis) decreases.

https://emj.bmj.com/content/20/5/453

The probability of correctly rejecting the

null hypothesis when it is false;