Elrond and Gandalf
 

At the end of the Hobbit, Elrond and Gandalf talk about how they kicked the "Necromancer" out of Mirkwood, but Gandalf says something about how he wishes he could destroy him for good. Elrond goes on to say that he doesn't think that will happen for a long time and he will be haunting ME for many years. What do you think about this? How come Elrond is always dispairing and never thinks they can defeat Sauron?

Sauron's existence was bound to the existence of the One Ring. As long as it survived, he would survive. Many people (including Gandalf) thought that the Ring had been destroyed after the Last Alliance defeated Sauron, but that was not so. Elrond knew that the Ring was still abroad, because he had been there the day that Isildur had refused to destroy the One Ring. It's a bit of foreshadowing for Lord of the Rings and the future struggle over the One Ring.

I do not think that Gandalf was ever under the impression that the Ring was destroyed after the Last Alliance. Saruman told him it had gone down the Anduin to the Sea and would be lost forever, but I do not remember any mention of the wise believing it had been destroyed.

Perhaps because Elrond saw as a child the slain of his friends in the Mouths of Sirion, was separated beyond hope of his beloved twin who had chosen the path of mortality, the destruction of Eregion, the Last Alliance, the awakening of the Necromancer, the fall of Arnor and Angmar... Perhaps because him even more than Gandalf has seen the Shadow fall only to recover and return in a diferent guise.