The Categories tab displays a global, hierarchical, category tree, with a check-box next to each category name. Selecting a check-box 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 check-box. You won't be able to create categories here, but you can expand checked categories , all categories or you may collapse all categories using appropriate buttons.
To assign product(s) to particular category positions you can use the similar option in the top toolbar as well: