The median is the 50th percentile. Half the values are greater than (or equal to ) the median and half are smaller.
The confidence interval of the median is computed by a standard method explained well in Zar (pages 548-549), based on the binomial distribution.
•The confidence interval of the median is not symmetrical around the median.
•You do not need to assume that the population distribution is symmetrical in order to interpret the confidence interval.
•The confidence interval begins and ends with values in the data set. No interpolation.
•Even if you ask for 95% confidence level, the actual confidence level will usually be different (especially with small samples) and Prism reports this.
J.H. Zar, Biostatistical Analysis, Fifth edition 2010, ISBN: 0131008463.