Depending on the rights allocated to your user account you have multiple options for approving reports.
When a report needs to be approved by someone else after your approval (for example the finance department) you can use the 'Approve and forward option'.
When you choose this option a dialogue box will open and you can see the email address of the next approver and you can add a message for the next approver.
When you select 'Approve' the report will get the label 'processing' and is forwarded to the next approver.
If you are the last approver in the workflow you can choose 'Final approve'. When you choose this option a dialogue box will open and you can add a note to the report. It is also possible to notify all previous approvers by checking the box. The owner of the report is always informed.
When you select 'Approve' the report will get the label 'Approved'.
More information on the status of a report can be found here.