The confidence intervals (limits) for the mean give us a range of values around the mean where we expect the "true" (population) mean is located (with a given level of certainty, see also Elementary Concepts). In Basic Statistics you can request confidence intervals for any p-value; for example, if the mean in your sample is 23, and the lower and upper limits of the p=.05 confidence interval are 19 and 27 respectively, then you can conclude that there is a 95% probability that the population mean is greater than 19 and lower than 27.

More precisely, if you repeatedly calculated this interval from many independent random samples of the same size, then 95% of the intervals would, in the long run, correctly bracket the true value of the mean, or equivalently you would in the long run be correct 95% of the time in claiming that the true value of the mean is contained within the confidence interval. Thus, technically, the 95% refers to the procedure for constructing a statistical interval, and not to the observed interval itself (see Hahn & Meeker, 1991, p. 31).

If you set the p-value to a smaller value, the interval would become wider thereby increasing the "certainty" of the estimate, and vice versa; as we all know from the weather forecast, the more "vague" the prediction (i.e., wider the confidence interval), the more likely it will materialize. Note that the width of the confidence interval depends on the sample size and on the variation of data values. The calculation of confidence intervals is based on the assumption that the variable is normally distributed in the population. This estimate may not be valid if this assumption is not met, unless the sample size is large, say n = 100 or more.