Dragon Story is a simulation game where player can look after, buy, breed and evolve a variety of dragons.
Dragon Story is similar to other "social" games in that you mostly watch (or don't watch) progress bars when things happen. You need to hatch dragons from the nest, build a habitat for that type (red dragons, green dragons, etc) and then feed your dragon to level it up. Dragons make money over time, so you need to periodically click the "collect" button on the habitat so it doesn't fill over. You make food on the farms, which also takes time. When your dragon has hit a high enough level it can evolve and make more money.
You also get quests to do these things and teach you the basics, which give YOU experience points and cause you to level, unlocking new dragons and items.
You can also place flowers for whatever reason, and there is a social aspect.
Since Dragon Story is a free-to-play game, you can quit watching progress bars by using gold bars, which you get by spending real money. A lot of itmes in game also cost gold bars, such as every type of dragon after the first three.