Piercings allow you to add some more power to your gear, as well as some customization. In order to pierce items, you must use Moonstones (obtained through killing monsters, chance boxes, or various rewards), and Scroll of GProtect (purchasable from Gisela in Flaris).
Both weapons and armor (suit) can be pierced, but the cards used for weapon and armor piercing are different. Once you've pierced an item, you can double click an appropriate card, then click the item to apply the card to the piercing slot.
You can pierce your items, as well as remove pierced cards by visiting BoboChan in Flaris.
Suits can be pierced up to 4 times, while weapons can be pierced up to 10 times. Unlike Monster Piercings, you do benefit from normal piercings in offhand weapons, like shields or a sword/axe (for blade/slayer).
Monster Piercings use Monster cards to give your weapon unique boosts. They're categorized into three tiers of rarity: General, Rare and Ultimate. General and Rare cards can drop from monsters in Madrigal, while Ultimate cards can only drop from bosses of the same name (for example, the [Ultimate] Clockworks Card drops from the Clockworks boss).
In order to use Monster Cards, you must Monster Pierce a weapon with BoboChan using Monster Stones (which drop from monsters) and Scroll of MProtect (purchable from Gisela).
4 Monster Card Piercings can be obtained with Monster Stones, while the 5th Piercing can only be obtained through Enhanced Monster stones, which you can obtain through participating in the Hunter Games.
Some additional restrictions on Monster Card Piercings:
You can only have 1 Ultimate Card per weapon
You can only have 2 Rare Cards per weapon
You only benefit from Monster cards in your main-hand weapon