I found this in the Augur documentation:
Once the event underlying an Augur Market occurs, the Outcome needs to be determined in order for the Market to Finalize and begin Settlement. The Outcomes of events are reported by Augur’s Decentralized Oracle, which consists of profit-motivated Reporters reporting on the Outcome of events. Anyone who owns REP tokens may participate in the Reporting process by submitting a Designated Report or First Public Report, or by Disputing Tentative Outcomes. Reporters whose Reports concur with consensus are financially rewarded, while those whose Reports do not concur with consensus are financially penalized.
So there are 3 options to participate in reporting:
designated reporter: chosen by the market creator, submits their answers to the market while staking REP
first public reporter: the designated reporter fails to submit an outcome, and the market moves to the open reporting phase. any user can be a first public reporter, they submit their answers while staking REP.
disputing: any user can challenge a tentative outcome by staking REP and using a dispute bond.
There should be more details in the docs on how to actually do each of these things (I'm not sure if they have actually been developed yet, so I can't answer your question as specifically as I'd like to).