Hell exists for those whose sin is not covered by the atonement of Christ. One should note, as per the answer to the earlier question of what makes Jesus’ death significant, that the only ones in Hell will be those who would not consent to having the merit of Christ’s life applied to them.
The two objections that immediately come to the fore concern why those who persist to the end in unbelief are not simply annihilated, or why, if they cannot be annihilated, they are ever created to begin with. With respect to the first, it is indeed the case that those in Hell cannot be annihilated. This is because they exist by virtue of participation in God’s being; in other words, their being is but a function of God’s being, in the sense that they eternally reside in his power to sustain in being (if they didn’t, they could never exist). But since goodness applies to the whole of God’s being, it follows that whatever resides within his power to sustain is good as well. But if that is true, then annihilating what now exists would constitute the annihilation of a good, which a good God would and could never do.
With respect to the second, the objection assumes that (at least in some cases) non-existence could be better than existence. But non-existence is not anything at all, and thus is not something that could be better than anything. And if it is not true that non-existence is ever better than existence, then it is not better to be annihilated than to be in Hell.