Achievement Item Rewards
Achievements primarily award you points. Sometimes you will receive an additional reward, such as a special Title or a unique item. In order to actually receive the item rewarded after completing a qualifying achievement, you'll need to travel to to Old Gridania and talk to Jonathas at Apkallu Falls and select the Share a tale dialogue option.
Here is a complete list of the Achievements which reward items. There are currently 113 rewards.