When using the word "department" after the name of a department, does one capitalize it? For example, "Advertising Department?"
When using the word 'department' (or 'group', 'committee', and the like) as part of a recognized name it would be capitalized:
Bill recently joined the Advertising Department.
If using 'department' not as part of a specific name it would be lower case:
We had a party to welcome Bill to the department.
As a further note, you could also drop the 'Department' and re-write the first sentence as:
Bill recently moved to Advertising.
But if using 'advertising' not as the name of a specific department, but rather as a field of expertise, you would write:
Bill works in advertising.