All the tickets in your system will be assigned to a department, and that department will be within a brand. If your brands are your different products, your departments for those brands could be the different types of ticket you'd get for those products - we've added Sales Enquiries and Technical Support as default departments, but you can add, edit & delete these as you wish.
Whereas brands can be created independently, departments require a couple of things to be in place beforehand. Firstly, a brand: all departments must belong to a brand, even if it's just your generic company one (see above for how to create brands). Secondly, an escalation path. Escalation paths are explained more in depth here, but essentially they define which members of staff will be assigned to the tickets in that department.
On the Departments page (fifth link down in the General Setup section of the Admin navigation) there'll be a list of all your current departments, grouped according to the brand they've been placed within. To add a new department, click the green button labelled 'Add New Department'. Select the brand you'd like the department to belong to, type in a name for your department, select the escalation path, and decide whether or not you'd like this to be a private department, or one you're happy about your customers seeing the tickets within and using in a public interface. Then press 'Create Department', and your new department should appear in the list, under the brand you assigned it to.
To edit your departments (most likely, the escalation path assigned to them once you've organised your staff and teams), just click the department name or the pencil icon on the right.
Deleting a department can be done by clicking the red 'X' next to the pencil icon, but you must first ensure that the department does not have any tickets assigned to it; tickets must belong to a department.