How can I interpret a 100(1-alpha)% confidence interval?

We often refer to a confidence level as the probability that a specific parameter will be contained in a given interval. For example, when we fit a 95% confidence interval to a fitted line, we say there is a 95% probability that the "true" fitted line (in the population) falls between the interval. As Hahn & Meeker point out in their book on statistical intervals (Wiley Series in Probability and Mathematical Statistics, 1991), this definition is common, but not entirely precise:

A 100(1-alpha)% confidence interval for an unknown quantity Theta may be characterized as follows: 'If one repeatedly calculates such intervals from many independent random samples, 100(1-alpha) % of the intervals would, in the long run, correctly bracket the true value Theta [or equivalently one would in the long run be correct 100(1-alpha)% of the time in claiming that the true value of theta is contained within the confidence interval].' More commonly, but less precisely, a two-sided confidence interval is described by a statement such as 'we are 95% confident that the interval theta-lower to theta-upper contains the unknown true parameter value of theta.' In fact the observed interval either contains theta or does not. Thus the 95% refers to the procedure for constructing a statistical interval, and not to the observed interval itself.