The Memory Core is a part of a person's mind that holds all of a person's memories. If a memory is removed from a Memory Core, it will be forgotten. It is possible to "enter" the bubbles to see the memory of the person and to remove the memory from the core to the real world. Once in the real world, it becomes a real object and can be destroyed easily.
Each memory seems to picture an object within a sphere from that certain memory, such as how the cherry blossoms relate to Marceline and Ash's breakup. In Marceline's memory core, pictures of the memories included cherry blossoms, a five-fingered hand, a wolf, a snowman, a shovel, a grave, a tree, an aquatic creature, and a candle.