A family pot is a pot that sees multiple players at the table going to the flop. In the strictest of sense, a real family pot is the one where all players pay to see the flop, but the expression is commonly used to describe pots where five or six out of nine players get involved.
For example, you raise from under the gun, and the player next to you calls. The next player to act folds, and then three more players call. It folds to the big blind, who throws in the calling chips, announcing: we have ourselves a multiway pot.
Despite a friendly-sounding name, family pots can be hard to navigate and can lead to some very tricky situations. Ranges in these situations are wide and not very well defined, so players sometimes wake up with the strangest of hands.