This is the best of all games wrapped up into one.
I am a 30 yr old parent of 3, and I am a moderator on a Minecraft multi-player server. This game is great for all ages, as there are many, many ways to play, most of which have been addressed already. But I am in a unique position as a multiplayer server moderator and a parent, so I hope you will give my review some consideration.
Small children can play in "creative mode" which is basically infinite Legos on a beautiful landscape. "Survival mode" (age 7-10) provides an adventurous scenario where there is just enough of a sense of danger to kick-start your motivation to quickly learn to follow crafting recipes and develop basic tactics and problem solving strategies.
Playing on a multi-player server (age 10+) allows you a collaborative experience with people all over the world. Yes, sometimes there are conflicts on a multiplayer server, but on most servers, the moderators are there to ensure safe and fair gameplay, and there are opportunities for children to learn responsibility and real world problem solving skills. It is very important to choose a server that makes your childrens' safety a priority.
Once your child is knowledgeable and comfortable on a Minecraft server (age 12-18), a whole new experience begins to unfold. For example, on the server I help moderate, there are opportunities to create your own town and learn people and project management skills, and participate in economic and political systems on a large scale. For players that follow the rules and show initiative and loyalty, there are opportunities for advancement to staff positions where they are trained to help other players, develop self-confidence, and become role models. Eventually a player may be able to earn an internship by learning technical skills and help customize the game itself by assisting with the development of plugins that modify the way the game works on the server. They may learn valuable, transferable skills for website and app development, as well as large scale personnel and project management.
The main things parents should be concerned about is the quality of the multiplayer servers and how well moderated they are. The parent should spend a few hours playing on the server and studying the server's website before allowing a child younger than 12 to play on it. There is also the concern that the child will become obsessed with this game, because it is so awesome and intellectually stimulating! As with any form of entertainment, it is very important to set clear boundaries.
I hope that parents can realize how useful this game can be in many aspects of their child's development, and how fun it is for all ages, even parents!