A user’s public display name is what is display as the “author” if you choose to use that field on articles, instead of a byline.
This can be a mix of their first and last name, or you can set a “nickname” which can really be whatever you want it to be.
From there, you should choose from the “Display name publicly as” dropdown, the name you want to be displayed on the site.
Remember to click “update user” at the bottom of the page to save your changes.