A Round is an instance of a stage. Each time a proof enters a new stage, it also enters a new Round.
Round Diagram for an Approval Manager Approval Process
Information on activities that take place and events that occur is saved in the Round information, ensuring that a complete history is kept and made available as necessary to authorized users.
In the capture of the Round diagram above, we can see that a file went from the "Copyread" Approval Stage back to the Production Stage (presumably to fix a text error). From the Production Stage, the file went to the "Customer" Approval Stage, and from there it was sent to the Finished Stage.