I keep reading about the need to check for autocorrelation in MCMC. Why is it important that the autocorrelation is low? What does it measure in the context of MCMC?
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Autocorrelation is a measure of how much the value of a signal correlates to other values of that signal at different points in time. In the context of MCMC, autocorrelation is a measure of how independent different samples from your posterior distribution are – lower autocorrelation indicating more independent results.
When you have high autocorrelation the samples you've drawn don't accurately represent the posterior distribution and therefore don't provide as meaningful information for the solution to the problem. In other words, lower autocorrelation means higher efficiency in your chains and better estimates. A general rule would be that the lower your autocorrelation, the less samples you need for the method to be effective (but that might be oversimplifying).
First, and most obviously, if autocorrelation is high then N samples are not giving you N pieces of information about your distribution but fewer than that. The Effective Sample Size (ESS) is one measure of how much information you're really getting (and is a function of the autocorrelation parameter).
Relatedly, autocorrelation gives you unrepresentative samples 'in the short run'. Moreover, the more autocorrelation there is, the longer that 'short run' is. For very strong autocorrelation, the short run might be a good fraction of your total samples. The usual direct remedies are re-parameterisation or sampling parameters that you expect to be intercorrelated in blocks rather than separately since they will otherwise generate autocorrelation in the chain. People often also 'thin', although there is some discussion about how useful this is in solving the underlying problem, e.g. here. Kass 1997 is an informal discussion of the issues, though there's probably something newer that others can recommend.
In short, a strongly autocorrelated chain takes longer to get from its starting conditions to the target distribution you want, while being less informative and taking longer to explore that distribution when it gets there.