While players of Xbox 360 and PlayStation 3 have to choose between Achievements and Trophies, in Clash of Clans you get both! Achievements in Clash of Clans more closely match the name on Microsoft's console. The menu for this is located below the trophy counter.
Achievements are essentially rewarded for completing certain challenges set by the developers. You can see those that you have already completed, as well as the ones yet to check off, in the achievements menu, which is accessible by clicking on the villager button on the left-hand side of the screen.
When there is a number next to this villager, it means you have an unclaimed reward to collect. Tap it and scroll through the list looking for the 'collect reward' option. This will give you a certain number of Gems for completing the achievement.