If you've gotten this far, you probably already know that a Department is the entity used to assign ownership of assets within the data center. You will still need to know who to call when there is smoke coming out of the server in Cabinet AA17, Position 4, and that is done by assigning Contacts to a Department.
First of all, to assign Contacts to a Department, you first have to pull up the Department Management screen. Under the details for that Department, you'll see a button to Assign Contacts. Click that to get a picklist of all possible contacts - any of the currently unassigned contacts are listed on the left hand side, and currently assigned ones are on the right. Adjust the boxes as needed and click submit.
It is perfectly reasonable and possible to assign a Contact to multiple Departments. Simply assign them as needed.