If you're talking about live chess.com tournaments (rather than OTB) I think everyone plays in each round until the tournament is over.
If you are talking about correspondence chess.com tournaments, go to the tournament page and scroll down. Click 'Pairings' on the left side of the screen. The top players in each group, with green arrows next to their names, will be advancing to the next round.
I have played in 3 or 4 tournaments and it's been confusing as to when my play is completed. In one of the tournaments I was waiting around for the next round and stayed connected but was looking around the site and then I find I'm listed as 'null', which I guessed was because I wasn't there when I was informed the next game would start. Is there a way to know whether you are still going to be playing another game?