I can honestly say that it can vary greatly depending on ship, department, managers, and CM involved.
I worked as YA on two different ships and both did them differently, and differently again depending on the managers we had at the time.
Basically, every CM is encouraged to have at least 1 'Magical Family', whilst within my department we kept a list so there were no repeats, that's not ship wide so its possible to be a MF for more than one department in a cruise.
If you were a MF for YA, the idea was that the kids/family would get an invite every night delivered to their room to certain events that the CM would think they would enjoy. They usually get at least one Magical Moment certificate and often get 'gifts'...
In YA we could have anything from 1-7+ MFs a cruise, and different CMs put a different amount of effort into it. As for becoming a MF, I would usually chose the families that I had 'fun' registering (ones that took their time, had a chat, had a laugh with, or those that were a bit nervous or worried about using the clubs) or if for some reason I hadn't done any registrations, the first kids that came up to me and played a games/had a chat/ did a craft etc on the first night.
The only way you can 'guarantee' being a MF again on another cruise is by meeting that same CM again on your next cruise and both of you remembering each other. I know a lot of CM (in my department) would select families who they remembered and who remembered them from previous cruises.
Most of the time they were pretty fun to do...