Hi!
So, first of all we should find a way to avoid nesting those lookups as they can cause excessive network traffic - can I assume that your Departments list is not a huge list? less than 2000 items? If Yes, we should collect that into a local collection during the App's OnStart or the first screen's OnVisible, like this:
ClearCollect(colDepartments, Departments);
This means that when our gallery is doing a lookup, it is looking at a local collection for each item and not doing multiple network lookups to the same list for each row.
Then your DepartmentName label can be the following, as I am guessing the error - if you hover over the red line on the equals - is that its a type mismatch, cannot compare type text to number or something like that, so assuming the People list has a Department ID of type Text rather than number:
LookUp(colDepartments,
'Department ID' = Value(LookUp(People, 'Employee ID'=ThisItem.'Employee ID').'Department ID')
).'Department Name'
Otherwise if it's the other way around and your Departments has a Department ID column of type Text:
LookUp(colDepartments,
Value('Department ID') = LookUp(People, 'Employee ID'=ThisItem.'Employee ID').'Department ID'
).'Department Name'