Very simple, the person will ask the next person below if they have watched or not watched the anime series mentioned. You can also reply with Watching, Plan to Watch, Dropped, or On-Hold depending on the person's status with that particular anime series.
9/22: For the manga/light novel variation of this game, click here.
Person 1: Shingeki no Kyoujin
Person 2: Watching
Person 3: Not watched
And so on...
Starting off with this: Great Teacher Onizuka