When you activate the "Allow users to sort categories" switch, this places an area on your index page above the category list where the visitor can select which field to order the list by, and a similar area over subcategories on the display articles in category page, and on the search categories results page.

When you activate the "Allow users to sort articles" switch, this places an area on your 'display articles in category' page above the articles list where the visitor can select which field to order the list by, and also on the search articles results page.

The visitor doesn't have to make a selection, it defaults to the order you've set at Admin -> Settings -> General. Still, you may prefer to switch off the option in order to simplify your site's visual style or to force a uniform result order for all.