You'll submit your expense reports to your manager or the hr/finance department, depending on how the workflow has been configured for your company. Different statuses offer you a clear indication on the workflow completion.
A report is "Open" will it has not been submitted yet. You will collect expenses during a specific period or event in an "Open" report, prior to submitting it to your manager. Once the period or event is passed, submit the report.
If you've submitted an expense report, the status will indicate "Submitted". The appointed approver will be notified by email that he or she is required to approve the report and (possibly) forward it to the next workflow stage.
If your report indicates "Processing", it means a certain workflow is being processed. It might for example entails that more than one approver needs to approve your report before being closed (and reimbursed). As soon as the first approver approves and forwards your expense report to the next stage, the report status will indicate "Processing".
Once your expense report gets "final approved", the status will indicate "Approved". Obviously, some details might be incomplete or you've made an mistake adding you're expense, for which your manager is able to reject the report. Why you're report is rejected will be shown in the report itself by your manager's comment.
Once an approved report is being or has been administratively processed, the status will indicate "Closed". Depending on how your company processes approved expense reports, this status may or may not be assigned. In most cases you may assume that reports indicating "Closed" will be reimbursed on your next payslip.