When I click Re-index site, and confirm, I get a message that the index will be indexed but it never does.
Run cron. (example.com/cron.php or go to Admin >> Reports >> Status and click "run cron")
You can also do it via drush to get around limit on the number of items that get indexed per cron run.
$ drush help search-index Index the remaining search items without wiping the index.
$ drush help search-reindex Force the search index to be rebuilt. Options: --immediate Rebuild the index immediately, instead of waiting for cron.