This item starts with no charges. Rather than gaining charges by clearing rooms or picking up batteries, Alabaster Box has to be charged by picking up Soul Hearts or Black Hearts – they would normally go towards Isaac's health bar, but instead go towards charging the box at a rate of one charge per half heart. It requires 12 charges – 6 full Soul or Black Hearts – to be fully charged.
If the box requires two or more charges and Isaac picks up a Black Heart, then the box gains charge as for a Soul Heart and the last Soul Heart (or half Soul Heart, if there is one) in Isaac's health will become black.
If the box requires only one more charge and Isaac picks up a full Soul Heart or a Black Heart, then the box will become fully charged and one half heart of the same type will be added to Isaac's health.
Soul health from Blended Hearts will also be used to charge the box. (They will, as usual, be used for red health first.)
Converter: Cannot use the charges stored in Alabaster Box.
Keeper / Tainted Keeper: As the Keeper and its tainted version cannot encounter Soul Hearts normally, the only possible way to charge the Alabaster Box is by having something that adds them directly to health, such as Schoolbag with Book of Revelations or Dream Catcher.
The item is likely a reference to the anointing of Jesus - while He was in Bethany, a woman came to Him with an alabaster jar of very expensive perfume, which she poured on His head as He was reclining at the table.
Picking up Bone Hearts while at 12 hearts will remove charges from the Box. The Bone Heart will not be added to the health bar, either.