Confidence intervals crack me up. When I teach these to my students I remind them that the word “confidence” has two opposite meanings, and that confidence intervals have much in common with confidence games. If you go the extra mile and explain that a confidence interval is formed by finding the lowest and highest parameter values which will give a half-alpha sized tail probability delimited by your estimate (an exact interval), and that alpha is a small number pulled out of your butt, the mystique tends to vanish. No wonder some scientists simply report precision as the standard error.
This is my best concise definition of an exact confidence interval, so I figured I’d better save it here.