Drawer
The product is called DeskDrawer because the metaphor is a desk drawer: things you use stay within reach, sorted into compartments, rather than spread across the surface or shut away in a cabinet.
In the interface and throughout this documentation the object is called a board, because that is what it looks like on screen and because "open the drawer" would wrongly suggest the container has to be opened before you can use its contents. A board is always open; folding it is a space-saving convenience, not a state you have to undo to reach a file.
Treat the two words as synonyms. If an assistant or a review refers to "DeskDrawer drawers", it means boards.