A Gemheart is a large gemstone, found within the bodies of all Greatshells. The bigger the animal, the bigger the stone. Even the smallest gemheart is worth a fortune and considered superior material for Soulcasting, offering near-limitless potential.
During the War of Reckoning between the Alethi and Parshendi, the two forces would often engage in combat over the gemhearts, a further testament to their worth. Although it is not specified why the Parshendi want the gemhearts, it may be speculated that they need them, amongst other things, to obtain food through soulcasting or by some other means.
"We won," Eshonai said, leaning back against the wall and folding her arms with a clink of Shardplate. "The gemheart is ours, we will continue to eat." 
Gemhearts found within chasmfiends encountered by Alethi warriors, were described as lumpy and uncut, pure emerald, in some cases as big as a man’s head.