There are a number of ways to buy Nintendo Stock:
Through a US broker with a presence on the foreign stock exchange.
Interactive Brokers have access to the Tokyo Stock Exchange, so you can buy Nintendo stock through them (ticker symbol: 7974)
Through a US broker that can trade the US listed depository receipt.
An unsponsored American Depository Receipt (ADR) trades on the Pink Sheets (ticker symbol: NTDOY). One should be wary of trading on the Pink Sheets as there is not a lot of liquidity and less strict supervision.
Through a US broker that can trade the foreign listing of the security.
For example, Nintendo is also listed on the Swiss Exchange (SWX or EBS) and the Fransfurt Stock Exchange (FWB).
Through a foreign broker that can trade the foreign listing of the security.
e.g. by opening a Japanese brokerage account.
Which one do I choose?
The ADR provides the advantage that they are usually listed in US dollars, which makes figuring out the capital gains when you sell the stocks a little easier (completing your schedule D of your tax return). However, the position will be subject to Foreign Exchange Risk - if the rate between the Japanese Yen and the US Dollar changes, the value of your position will change irrespective of how well the company is doing.
The primary exchange listing is usually the best, if your broker has access to that market, as they tend to have the highest transaction volumes and are therefore the most liquid, though you should consider the transaction costs, and the cost of purchasing the Japanese Yen you need to buy the security etc.
Secondary foreign exchange listings - These incurring two different FX risks. One to convert USD to Swiss Francs or Euro, and another to convert Swiss Francs or Euro to Japanese Yen.
Foreign brokerage accounts may be difficult to set up (e.g. there could be a language barrier), may have local regulations, and may not be as competitive in terms of price as US brokers, but it is still possible.
Consider the volume, liquidity, transaction fees and prices on each exchange. As a general rule, the primary exchange is the most liquid and highest volume.
That is the presence of other market participants willing to buy and sell at competitive prices. You can get a feel for the liquidity by looking at the spread (percentage difference between the bid and ask prices) and the volume traded. If a market is illiquid, you may not get the best price when you try to buy or when you try to sell.