Determination (DT) is like a will to live, it's what lets the soul live on after dying. It's also what makes Undyne the Undying so powerful. Humans have DT, monsters have a trace amount. Monsters that gain too much DT will stay alive, but they'll... I don't want to spoil the pacifist route.
Anyway, the person with the highest DT has the ability to load/save/reset. DT is also what makes the soul not break when your HP is 0.