They're parallel worlds.
In this game, Ichi no Kuni (First World) is Oliver's world with the old US town, and Ni no Kuni (Second World) is the fantasy world with magic and monsters and all. There are even more parallel worlds mentioned, as the place where some sidequest monsters originate from (and a sidequest in the PS3 version even shows one, which is basically a small zone from Hotroit/Motorville with different NPCs).
Ichi no Kuni and Ni no Kuni have direct influence on each other, notably the fact that each living being has a soul-mate on the other side, sharing a soul with him. Meaning, if a certain someone is ill ("hollowed"/"brokenhearted", where he's missing some emotions), you need to find his soulmate on the other side who will probably have some problem, and fix him up. This implies that whenever someone dies, his soulmate does as well as long as their connection isn't severed... and then there are certain people without soul mates. Oliver's search for his dead mother soul mate is what sets the entire plot into motion.
Ni no Kuni humans were forbidden from visiting Ichi no Kuni because the technological shock was deemed dangerous, and it was restricted so that it's only possible to people who know magic. People who know magic in Ni no Kuni were persecuted by the big villain, and Ichi no Kuni folks don't know magic. How Oliver got to know magic while in Ichi no Kuni is a late-game plot point.