Only Super users can add new values such as Department and Relationship. This prevents these values being added randomly and keeps the assignment of tags and categories clean.
If you want to add a new value to any of the following, you'll need to be a Super User:
People: department
People: relationship
People: tags
Organisation: type
Organisation: relationship
Organisation: tags
Adding New Values
To add a new value, such as a Person's Department, just click and choose the Add New value at the bottom of the drop down.
Or with tags, click in the tags box and type your new tag in lowercase:
Tags can be separated by commas or a carriage return. As such, commas are the only character you can't put in a tag.