Say I want to estimate the mean of a population.  In order to quantify the amount of uncertainty about my guess, I construct a confidence interval.  So you might see a statement like “A 95% confidence interval for the mean is (17.6,24.6).”

This means that 95% of the time the true mean is in the interval.

I just lied to you.

In the classical statistical framework, the true value of the mean is unkown and fixed.  Therefore, the probability that the true mean is in that interval is either 0 or 1.  It’s either in the interval all the time or never in the interval because the true mean is fixed.  There is nothing random about it.

What is random is the sample.  So, the true interpretation of a confidence interval is this: “If the confidence interval were consturcted in the same way over and over again, 95% of these similarly constructed intervals will contain the true values of the mean.”

So go forth with confidence in your statements about confidence intervals.

Cheers.