Probably the most common means of moving a task forward, is assignment. Assigning a task is the way to designate a user or group of users, as those responsible for the fulfillment of a given task.
The base Task [task] table (and therefore, all tables which extend it) contain two assignment fields: Assigned to [assigned_to], and Assignment Group [assignment_group]. If you've enabled Visual Task Boards, then you'll also have the Additional assignee list [additional_assignee_list] field.
Assignment drives a lot of default notifications, such as alerting the assignee when work notes are added by another service desk agent, or when the customer leaves comments. It's usually a very important part of the task-handling process.