Like the SEM, the confidence interval also quantifies the precision of the mean. The mean you calculate from your sample of data points depends on which values you happened to sample. Therefore, the mean you calculate is unlikely to equal the overall population mean exactly. The size of the likely discrepancy depends on the variability of the values (expressed as the SD) and the sample size. Combine those together to calculate a 95% confidence interval (95% CI), which is a range of values. You can be 95% sure that this interval contains the true population mean. More precisely, if you generate many 95% CIs from many data sets, you expect the CI to include the true population mean in 95% of the cases and not to include the true mean value in the other 5% of the cases. Since you don't know the population mean, you'll never know when this happens.
The confidence interval extends in each direction by a distance calculated from the standard error of the mean multiplied by a critical value from the t distribution. This value depends on the degree of confidence you want (traditionally 95%, but it is possible to calculate intervals for any degree of confidence) and on the number of degrees of freedom in this experiment (N-1). With large samples, this multiplier equals 1.96. With smaller samples, the multiplier is larger.
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