The "Categories" tab displays a global, hierarchical, category tree, with a checkbox next to each category name. Selecting a checkbox will associate this product with that category, and you can also select as many associated categories as you wish. For the categories to display on the homepage you must enable the root catalog checkbox. You won't be able to create categories here, but you can expand checked categories , all categories or you may collapse all categories using corresponding buttons.
To assign product(s) to particular category positions you can use the similar option in the top toolbar as well: