There isn't really a "family games" genre, but some games fit the description better than others. And we aren't talking Monopoly—21st-century offerings are much more sophisticated and less likely to result in bloodshed or estrangement. Typically, a family game is one appropriate for all ages, yet interesting and challenging for everyone, like Sequence or Set or Uno.
Sometimes family games are more hands-on, like Jenga (in which players pull pieces from a tower whithout letting the whole thing come crashing down). Others test players' knowledge (Cranium), logic skills (Forbidden Island), or teamwork (Pandemic). Some are just ways to pass the time, like the frustratingly popular Yahtzee.
A lot of the new strategy games make pretty great family games, too. Settlers of Catan might be over the heads of 6-year-olds, but most of the family will enjoy discovering and taming a new land. Carcassonne is even more accessible, and because games are typically short, younger players potentially have more chances to improve their strategy and problem solving abilities.
Of course, the greatest family game of all time is Dutch Blitz. It combines the aggressiveness of the German people with a card game played in real time without turns, can lead to considerable strife, and is more addicting than anything you've ever been addicted to. Basically 4-way solitaire in which the fastest player is the winning player, no game collection is complete without Dutch Blitz.