Toyota is listed on the Nikkei. The thing at the NYSE is called an American Depository Receipt (ADR). Basically, it is a coupon that says "Entitles the bearer to two shares of Toyota common stock in Japan."
In principle, the two do not necessarily have to move in lockstep. However, because of the massive size, prominence, and stability of Toyota, one would ordinarily expect them to correlate extraordinarily heavily as anything else is an exploitable arbitrage opportunity.
Various algorithmic traders play on these dual listings. Put a computer in between (in network terms) and you can trade ahead of the people in either japan or new york.
and of course you can also do regular pairs trading (if they drift farther apart).
In principle, the two do not necessarily have to move in lockstep. However, because of the massive size, prominence, and stability of Toyota, one would ordinarily expect them to correlate extraordinarily heavily as anything else is an exploitable arbitrage opportunity.