Assignments are pages, components, and assets that are in your possession or in the possession of a group to which you belong. This means that you can work on them, though some action may still be required before you can perform that work.
The Assignments box gives you an unfiltered number of items that belong to you. Clicking the box loads the Assignments pane from the Utility Bar. You can use the Assignments fields at the top to filter and search your items.