Armory tokens are an in-game currency that can be spent on purchasing Chests of Equipment on the Chests tab at the Market.
There are several types of Armory Token: Armory Token V, Armory Token VI, Armory Token IX, and Armory Token X. All these currencies are used to buy Chests of Equipment of the corresponding level.
For instance, Armory Tokens V can be used to buy Chests of Equipment III–V, containing items of levels III through V. Armory Tokens VI are used for purchasing Chests of Equipment VI–VIII, containing gear of levels VI through VIII; and so on.

You can start earning Armory Tokens at early stages of leveling up your profile in various game modes (Tournament, Challenges, Campaign, etc.), but the chests become available later on:
Chests of Equipment III–V: Level 20
Chests of Equipment VI–VIII: Level 50
Chest of Equipment IX: Level 80
Chest of Equipment X: Level 80

To find out where you can obtain a specific type of Armory Token, tap the corresponding chest, then the currency icon in the top right corner. You'll then see a tooltip explaining how to earn the selected Armory Tokens.