"Green card" is the nickname given to the United States Permanent Resident Card. Originally the card was green, hence the name. The current version of the "green card" is no longer green. It bears the card holder’s photograph, fingerprint, and other information, and it entitles the cardholder to permanent residency in the United States.
In order to obtain a green card, one must go through an application process. For the most part, the cards are given to people who are working in the United States, or to those with relatives there. Once you have a green card, you can use it, along with the passport from your country of origin, to enter the United States. You also do not need a visa to enter the United States as long as the card is valid. In order to maintain validity, you must visit the United States at least once a year, and stay there for a certain length of time. While you are accorded most of the same rights as an American citizen, you are not allowed to vote or run for public office in the United States.
Currently, Taiwan’s laws forbid people with citizenship in another country from running for president. However, the Central Election Committee (CEC) has said that those laws do not affect people with American "green cards". That’s because the cards give the holder permanent residency, not citizenship.
The opposition Democratic Progressive Party disagrees, saying that people holding foreign permanent resident cards should not be allowed to run for the presidency. There have been attempts to amend the law to ban people holding foreign permanent resident cards from holding public office, but all the attempts have failed.