"When Kronos created the abyss, he made sure that the key to opening it was a paradox, the power of a god coupled with the innocence of a child."
The Abyss of Tartarus was a cavernous maw beneath the Tapestry of Time on Mount Olympus. Hera told Hercules that the spirit of Kronos was trapped there until the end of time. (HTLJ "Reunions") Later, when fighting with Hercules, Hera fell into the Abyss. (HTLJ "Reunions")
The Abyss seemed to affect the memory of any a individual trapped in it, though the exact mechanics of this were unclear. Olympians trapped within were robbed of all their memories unless they came into contact with their thread from the Three Fates' loom, while titans who had been trapped had blurry memories.(HTLJ "Full Circle")
Individuals could only be released from the Abyss by beings who had both the power of a god, but the innocence of a child (something that virtually every god lacked). This essentially meant that only very young gods (or very young demigods of extreme power) could release someone imprisoned within. Presumably, it was not possible to coerce someone fitting this paradox to release a prisoner (since the coercion would ruin their innocence).
Zeus was able to free Hera from the Abyss of Tartarus by manipulating his grandson Evander into doing it. He was able to do this by using a game invented by Hercules to make Evander feel compassion for his trapped grandmother. Unfortunately, Evander also felt sorry for the titans trapped there as well, and unwittingly released them upon the world.(HTLJ "Full Circle")