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Subordinate is responsible for managing his/her own documents (timesheets, travel expense documents) as well as sending them for approval. A subordinate is under the command of a superior and, thus, takes the lowest level in the chain of approval with limited rights.
Coordinator has a higher level of control comparing to a subordinate. Each Coordinator has at least one Subordinate who is supposed to submit documents for approval. The documents are required to be verified and approved. In case the submitted document does not contain all the necessary information or is outdated, it can be rejected back to an originator (Subordinate) for editing or cancelling. A Coordinator may be a subordinate as well managing his/her personal timesheets and sending them for approval to a Coordinator of a higher level (or to an Approver).
Approver is responsible for approving documents and has the highest level of control for any department. Only approved timesheets are then sent to the Payroll module for further processing and payment. An approver can also be a Coordinator but cannot be a Subordinate.
For more information on how to set up the Chain of Approval, see This Chapter.