There is no kids club at BW...used to be but not anymore. There is one at BC, WL, Polynesian and perhaps AK. We have used the Neverland Club at the Polynesian several times and my dd loved it. In fact, she was depressed when she got too old for it!!! They take kids from 3-12...at least that's how it used to be.
The kids clubs are held in various resorts..see above...in a room apart from other areas. They will give the kids a meal or snack, depending on timing. There are various activities for the kids, depending on age and interests. At the Neverland Club, it was a Peter Pan theme. My dd told me there were limited video games, books, games, craft activities supervised by CMs, toys (both for girls and boys) or just areas for 'quiet time'. It's not cheap but it's a great thing.