Dungeons & Dragons is a pen-and-paper (PNP) role-playing game, where players create persona's or characters using rules, dice and some imagination. Similarly the game is played this way - there is a game master that is in charge of organizing and running a game or campaign and controlling the monsters or NPC's (non-player-characters) and the players that are in charge of their own PC's or Player Characters.
The name 'Dungeons & Dragons' (D&D) is a specific set of rules for a role playing game (RPG). There are other PNP RPG's for all kinds of settings, including sci-fi, modern world and more. Some games are fairly simple, having minimal rules, minimal detail, while others can be extremely complex. D&D is in-between, it can be played using only basic rules, but is also able to scale up and you can utilize the many available books for additional rules, game features and detail.
Once the rules are understood, it can be an enjoyable experience for years to come. It is something that I thoroughly enjoy, and most kids enjoy dragons, monsters, heroes, castles and so on. They especially enjoy using their imagination to make up stories and 'play'. However, children under the age of 12, or in my case age 7, do not have the patience to learn and follow even the basic rules for a game such as this.