The Skull Heart was an artifact from the game Skullgirls and influential antagonist in the universe. It had the ability to hear the desperate wishes of a girl or woman; depending on whether the wish was considered "pure" or not, she who wished would become a Skullgirl.
The Skull Heart worked with women and only women; this included anyone who identifies as female. It wouldn't respond or grant wishes to men (with one peculiar exception). Its personality caused its victim to desire the destruction of the world. Despite claims of it only creating a Skullgirl based on wish intent, all wishes made on the Skull Heart would make the wisher become a Skullgirl; this is because "wishes are selfish by nature".
Skullgirls were controlled by the Skull Heart itself. When it or the current Skullgirl was destroyed or killed, it would lay dormant and reappear seven years later. If a Skull Heart were to be left alone untouched, it would resurrect the current Skullgirl until she was put down again. Depending on the frequency of selfishness of the wish given to it, a new Skullgirl could take from the entire seven year limit to a few milliseconds to form from the one who wished.
It was possible to "overlay" the magic of the Skull Heart with other sources of magic. This was only touched upon once in NIC canon during the Magical Skullgirl arc.
Though its initial creation and appearance is vague, the Skull Heart is believed to have been made by the Trinity. It
has been used multiple times to grant wishes (and create Skullgirls as a result). In NIC, the Skull heart has appeared multiple times due to timeline resets and other factors. As a result, multiple characters have became Skullgirls and the artifact has been destroyed just as much. It was finally destroyed one last time after the Apocalypse, and with the Trinity either dead or missing, there's presumably no possibility of a new one being created. Per word of god, it's never coming back.