Destiny is a mish-mash of genres. Kind of divided into levels, but kind of not, you'll spend most of your time taking on missions from a central hub, teleporting to a planet, and blasting your way through some ancient ruins, a giant cave, or some other otherworldy structure. With little in the way of a story tying things together, there's little motive behind each quest - instead, you've just got to shoot some things, get to the end of a level, and rinse/repeat. Destiny's missions are stratified according to their difficulty, so it's quite flexible to casual players and those completely new to shooters; you simply pick the difficulty that's appropriate for you. The more you play, the better weapons and armour you'll find, and the more you'll level up, making your character more powerful, and opening up new levels.
If your child's used to first person shooters, they'll likely be in familiar territory here, although there are a few big differences. Perhaps the biggest is that co-op play is practically required to progress through the game, so you'll need to team up with randomers - or friends - to get past certain tricky enemies or levels. Simply blasting your way through the game on your own isn't going to be all that easy, if it's possible at all. The only thing parents should keep in mind with this is that the game has no support for local, split-screen co-op. If you want to play with a friend, they'll have to be a friend, in a different house, with their own copy of the game, rather than one sitting next to you.
As Destiny is designed to be an online co-operative shooter experience, you have to constantly be connected to the internet, and as such, will constantly be running into other players going about their business. When going into certain missions, you'll also be automatically be matchmade with other players, but you won't be able to talk to them by default, without specifically adding them to your group (or "fireteam" as it's known in game). On one hand, that can make the game's world feel a bit quiet at times, but it does mean you won't have to end up having to talk/listen to people you don't want to.