Here's one that surprised me when I found out about it years ago, because I'd never really given it thought: There's a correlation statistic on columns for how well the values in that column match the row order on disk, which can influence a few different things.
In my case a query that retrieved a ton of data with an ORDER BY was using a sort and taking like two hours to run (a data source for an ETL process) - turned out because of a really bad correlation postgres was refusing to use the index, because the random access would be even slower, so it did a table scan then sort. After figuring this out and discovering the CLUSTER command (reorders the data on disk to match an index), it did an index scan and didn't need to sort at the end, was able to start streaming results immediately, and finished the entire query in like ten minutes.
Just a nice example of where the obvious "use query hints to make it use the index" would have been the worst option, instead figuring out why postgres didn't want to use it and fixing that resulted in something much better.