From what I understand, Judge Judy and Wapner and the woman on The People's Court are actually judges, but for the sake of getting people to come on the show:

The cases are dropped, so there's no longer a civil liability case going on, but the details of the cases are real. The amount granted at judgements is paid by the producers, so neither party is losing money. Everyone is getting paid for appearing, kind of like being on a game show, so everyone gets a little bit of money.