The Catalyst is the final piece of The Formula, without which it is useless. It survives by living inside of a host body. When the host dies, so does the catalyst. Those with the Catalyst seem to be able to transfer it to others at will (i.e., Ishi Nakamura transfering it to her son Hiro) but Arthur Petrelli was also able to forcibly take it from Hiro using his Power Theft ability.
Originally, Claire Bennet as an infant was to be given the catalyst, but Hiro Nakamura, following the images in the last 9th Wonders comic book, teleported Claire and himself to the day when Noah Bennet was given his assignment to watch Claire, and driven by his love for his lost mother, Hiro made himself known to her as from the future and Ishi gave Hiro the Catalyst instead of Claire and then died. It was almost immediately afterwards stolen from Hiro by Arthur Petrelli and he left Hiro to die hanging from a flag pole while teleporting Claire back to Angela Petrelli. Later, when Arthur was killed by Sylar, the Catalyst died with him.