Gems are the premium currency on the game, similar to all Supercell games, and can be obtained with real money through in-app-purchases or by opening Brawl Boxes.
They can be used to buy Brawl Boxes, Skins, Key Doublers, and Coins.
The fastest way to progress on the game is by using Gems to directly Boxes (especially the Mega Box).
But you can also buy Token Doubler, which doubles the number of Tokens you get from battles. This is currently the best cost/benefit of the game.
This type of purchase is quite interesting, as it feels like a “subscription” once you get used to it. If you want to progress faster, keep buying them every week.
It's worth noting that some good sales might show up at the Shop. The price of Token Doublers and Big Boxes can drop for this period. So far there is no schedule or a way to predict when these sales will show up.