Definition
A confidence interval is a range of values constructed from sample data such that, if the same procedure were repeated over many independent samples, a specified percentage of the resulting intervals would contain the true value of the parameter being estimated. A '95% confidence interval' is built by a procedure that succeeds 95% of the time in the long run.
The phrasing is deliberately about the procedure, not the specific interval — that subtlety is the source of most misinterpretations.
Why it matters
How it works
For a sample mean from a normal population (or a large sample, by the CLT), the 95% confidence interval is the sample mean plus or minus 1.96 × standard error. The standard error is the standard deviation divided by the square root of n, capturing how much the sample mean would vary across hypothetical repetitions.
Different parameters (proportions, ratios, regression coefficients) have different interval formulas, but the conceptual recipe is the same: pivot off the sampling distribution of the estimator to get a probabilistic statement about the procedure, then invert it into an interval.