Here's what you can do to get quite close to the display you want. I'm assuming that your 'Staff1', 'Staff2' etc are user accounts. If you have a content type 'Staff' and your 'Staff1', 'Staff2' etc are nodes, the following could still be done, but some of the actual choices would be different. You didn't specify, so I'll assume they are actual user accounts.
Create a new view, but on the first page, make sure to select that this will be a listing of 'Taxonomy terms' and go ahead and select the exact vocabulary they should be pulled from (something like 'Assignment' in your case):
Under the 'Advanced' section, add a 'Relationship' to your view. If it is indeed user profiles that these terms are attached to, select the 'Taxonomy term: User using ' relationship. When configuring the relationship, you can select that you want that relationship to be required, if you want to only display users who have an assignment or you can leave it as unrequired if you also want to have a list of all users without an assignment as a group! This will be 'Taxonomy term: Node using ...' if you have a content type for Staff.
Finally, just add 'User: Name' to your view and in the table settings (under 'Format'), select to Group By your 'Assignment'. I'd also then check the 'Exclude from display' checkbox for the 'Assignment' term because it's a little redundant to display it as a column if you're already grouping by it.
In any case... that'll get you pretty close, though you can certainly do some extra css-ing to get it looking prettier.
Hope this helps and at least sets you on the right track...