For this achievement you must activate a Boost Gem 100 times in battle over time. A Boost Gem comes in 5 different categories, here is the list of each different category as well as how some of them aid you in battle. While some may be pretty straight forward such as Attack and Defense others may confuse players on what exactly they do:
Attack: Increase attack power
Defense: Increase defense
Speed: Speed increases your characters jumping and movement speed, the speed gems do not alter things such as the start up frames of your moves
Cross Gauge: The Cross Gauge gems increase the amount of Cross Gauge you build during your attacks and can reduce the cost of your Super and Cross Arts by up to 40% depending on the gem you use.
Vitality: Vitality Gems can gradually increase your characters health during battle or give them a set amount right away depending on the gem you use.
To equip Gems go to Customize from the main menu then Gem Unit. Only 3 Gems can be equipped per character so that means you can activate 6 gems per match towards this achievement. Easy gems to activate are usually gems that require actions from you such as hitting your opponent a certain amount of times which is the Attack gems. Look at all the activation conditions and use the ones you feel more comfortable with yourself to activate more often. Keep in mind Assist Gems from A Glimmering Light will not count towards this achievement as they are Assist Gems and not Boost Gems. This will unlock after the match you activated your 100th Boost Gem on.
To check your progress for this achievement, go to Player Data at the main menu. Select Player Records and scroll down to see how many "Boost Gems Activated" you have to keep track.