You can add a new category from any part details page by selecting the Category drop down menu, scrolling to the bottom, and clicking Add New.
You can also add new categories in Office > Back Office Settings > Table Editor > Categories.
This same page under Table Editor will display an edit and delete icon for each category. Please check for any parts that may be assigned to a category prior to deleting it, on the appropriate inventory page (i.e., Manage Products, Rentals, etc.).
Every category requires a department. Categories are used for organizing and searching for inventory and sales reports, whereas departments are used for reporting and accounting. Departments tend to be a more broad definition (ie, Apparel vs T-Shirts, Hats, and Gloves).