Joomla has two search options: Search and Smart Search. Depending on which you choose may dictate some factors.
For Search (Components > Search) you will need to manually enable gathering search statistics in the options. This is off by default, as it can accumulate massive amounts of data if there's lots of searches on a popular site.
Once you're gathering statistics, going to Components > Search will show you what people have searched for on the website.
Smart Search works similarly, but the main difference is that Smart Search creates a deeper index for your site. Again, turn on Gather Search Statistics in the Options.