I hear this question a lot from parents. As an educator I encourage parents to play games with their children because yes, fun games can be educational.
Let's look at some of the concepts that are reinforced by playing games. First, consider the fact that by playing games kids are learning to take turns. Children grow up with a natural need for instant gratification. Game play teachers patience and an understanding that everyone gets a chance and that their turn will come again.
Secondly, there are usually a set of skills required to play a game. The skill set may be very simple like moving a pawn around the game board. But even in this simplistic setting the kids are learning to associate numbers with their values as they count the spaces and move their game game piece forward. Some games reinforce knowledge of color while others include money skills. Some games will require skills that are obvious while others will be less so.
Some fun games go beyond basic skills and require higher level thinking like reasoning, understanding cause and effect, strategizing and synthesizing information in order to accomplish the object of the game. It is exactly these skills educators hope to instill in students. What better way for them to learn than to play games and spend quality time with their parents and families.
Whatever game you choose to play, I am fairly confident there are skills that young players will learn to reinforce those who learn in school. Be sure to have age appropriate games available and follow your children's lead. Most importantly, Have Fun!