Precursive Task Categories are colorful labels that can be assigned to Tasks and usually identify the type of work that the Task requires (i.e. 'Customer Call', 'Internal Meeting', 'Admin').

Categories can be assigned to Tasks by your Salesforce Admin or Superuser when creating Task Templates (About Precursive Templates) and you may see them appearing automatically when creating a Precursive Board using a Board Type. However, they can be also assigned manually in case you need to add a new Task to your Board.

You can assign a Category when creating or editing Precursive Tasks (Add and Edit a Precursive Task). The Category field is a picklist - you will only be able to select one entry among the Categories that have been created for you by your Salesforce Admin or Precursive Superuser. If you don't see a Category previously used, please check with them if that Category has been deactivated.

Create & Manage Task Categories 
Add and Edit a Precursive Task
About Precursive Templates