Adventures are limited time campaigns, accessed from the Battles or dailies tab. You have a limited pool of tokens which determine how many battles you can do per day.
If you want to grind this out, keep an eye out for an adventure that is repeatable that awards gems upon completion. Use the gems for completing the adventure to purchase more adventure tokens, rinse and repeat! Once you hit 10, the achievement will unlock.
In particular, keep an eye out for the Dreamhold, Khazdhuli, and Flametree expeditions, as you earn a top tier relic for completing these which are incredibly useful toward the endgame! (once Eveline is Mythic you can trade 2 top tier relics for one of your choice, which I've found is the easiest way to get relics to hone your best Mythic gear)
Note: if you're a longtime player before the 3.0 update, this will not unlock retroactively. It only counts adventures from after the 3.0 update.