What does the story of Noah's Ark foreshadow in the New Testament? I have seen Noah's Ark listed one of the "types" in Scripture -- something that occurred, but that also foreshadowed something else that would come later.

What exactly, then, does the story of Noah's Ark foreshadow?
The flood is explicitly used typologically twice in the New Testament.

Jesus, speaking of the final judgment (Matthew 24:37-41), says it will be like "the days of Noah".


  For as the days of Noah were, so will be the coming of the Son of Man. For as in those days before the flood they were eating and drinking, marrying and giving in marriage, until the day Noah entered the ark, and they knew nothing until the flood came and swept them all away, so too will be the coming of the Son of Man. Then two will be in the field; one will be taken and one will be left. Two women will be grinding meal together; one will be taken and one will be left.


1 Peter 3:20b-22 says that the flood prefigured the sacrament of baptism.


  God waited patiently in the days of Noah, during the building of the ark, in which a few, that is, eight persons, were saved through water. And baptism, which this prefigured, now saves you—not as a removal of dirt from the body, but as an appeal to God for a good conscience, through the resurrection of Jesus Christ, who has gone into heaven and is at the right hand of God, with angels, authorities, and powers made subject to him.


Over the centuries Christians have found additional typology in this story:


The ark has "rooms" for the animals (Genesis 6:14); Jesus says that his father's house has "many rooms" (John 14:2).
The ark had one door (Genesis 6:16); Jesus claimed to be "the door" to salvation (John 10:9).
God invited Noah to come into the ark (Genesis 7:1); Jesus said, "Come to me, all you that  are weary" (Matthew 11:28).
The ark was made of wood (Genesis 6:14) just as the cross was.
Because God saved him, Noah "remained alive" (Genesis 7:23); Jesus gives us eternal life (John 10:28).


This is only a sample; you can find longer lists here and here. Some of the connections are more explicit in the King James Bible; newer translations use different phrases that don't always match up, but the similarity should still be apparent.