What kind of game are Miss Leefolt and the other ladies playing at the very beginning of The Help?
The women are playing bridge. This is a card game that is often used to organize social occassions or provide an activity to serve as the heart of a gathering. Bridge clubs were common in the US for quite a while in the leisure classes.
The women in this novel qualify as being part of the leisure classes, even if they are not all wealthy, as they 1) all have hired servants and 2) they are able to play bridge in the middle of the day. Without having to do their own housework, these ladies are able to gather for social events and play cards if they choose to. This is part of the idea of "Southern gentility", which persisted into the Jim Crow era.