To have products appear on the website they must be assigned to categories.
Categories tab shows which categories the product is currently assigned to, if any.
Click the 'Edit' button to view the category browser where you can select one or more categories. Selected categories as shown as a tick on a green button. Unselected categories are shown to take on a grey button.
Only one category can be set as the home category. On the product detail page of the website the home category is the category that the product is shown to belong to. This depends on the template implementation, but typically the category link on the product page links to the home category.
To remove a product from a category tickets corresponding checkbox in the category list and click the remove button.