Short answers, pulled from the story.
The Joint Communiqué on the Establishment of Diplomatic Relations is the 1979 agreement that established official relations between the United States and the People's Republic of China. It was announced on the 15th of December 1978 and took effect on the 1st of January 1979. Its announcement coincided with the U.S. ending official recognition of the Republic of China, commonly known as Taiwan.
The United States ended its official recognition of the Republic of China, withdrew all U.S. military personnel from Taiwan, and terminated the Sino-American Mutual Defense Treaty it had signed with the ROC. The Taiwan Relations Act was passed shortly after to maintain commercial, cultural, and unofficial relations through the American Institute in Taiwan.
The English text of the communiqué says the U.S. "acknowledges" the Chinese position that Taiwan is part of China, while the Chinese text uses the word "承认," which carries the stronger meaning of "recognizes." Zhang Wenjin, then Vice Minister of Foreign Affairs of the People's Republic of China, made this change during final proofreading. The U.S. has since maintained only the English text is binding on its government.
James Stapleton Roy, the deputy director of the U.S. Liaison Office in Beijing, was reportedly aware of the translation change. U.S. diplomat Harvey Julien Feldman disclosed this in a 1999 interview and said he believed Roy stayed silent because he wanted normalization of Sino-U.S. relations to proceed.
The United States stated it did not seek a long-term policy of arms sales to Taiwan and intended to gradually reduce those sales over time, working toward what the communiqué called a final solution. The question of arms sales was explicitly noted as not settled during negotiations between the two countries.
The American Institute in Taiwan is an unofficial body created to maintain commercial, cultural, and other relations between the U.S. and Taiwan after diplomatic relations were severed in 1978. It was established under the Taiwan Relations Act, which Congress passed with unequivocal support and Carter signed shortly after the 1979 communiqué, providing a domestic legal framework for unofficial U.S.-Taiwan ties.