A Sintillian Heart was the living heart of a Sintillian. The word living is literal, as the heart is alive and has the ability to run around.
In "The Alpha Syndrome", a Sintillian came to the MiB to get agents to find his stolen heart. Jay and Kay discover the culprit is Alpha. Here, they learn that the heart is actually alive and runs around. However, Alpha shoots it. However, the agents are able to save the heart, and return it to its owner.