Why should we use 16-50-84 percentile instead of $\mu$ and $\sigma$ to characterize sampling results such as MCMC Can someone explain why we should use the 16-50-84 percentile rather than mean and standard deviation to characterize the average and uncertainties of the sampling results, such as MCMC (the Markov Chain Monte Carlo sampling)?
I look up the source code of a python package corner and I find they are using the 16-50-84 percentile. Can someone explain the difference between the 16-50-84 percentile and mean/standard deviation and when we should use which?

You can use either, and in fact I suggest you use both. The mean, median, standard deviations, and other percentiles all provide different information as summary statistics of the posterior. I'd also definitely recommend a histogram (or other density estimate) of posterior draws.
