AP Photo/Jorge Saenz, Pablo Martinez Monsivais, Files
The governments of China and Taiwan each claim to be the legitimate rulers of the entirety of China - well, most of the time. In 1949, China split between the mainland, ruled by Communists, and Taiwan, ruled by Nationalists. In 1992, the two agreed that there was one China. In practice, that's meant China leaves Taiwan alone, while Taiwan doesn't try to declare formal independence.
The current ruling party in Taiwan, the DPP, wants to declare independence. That means they won't always abide by the One-China niceties.
For the U.S. government, that requires being careful about things like taking phone calls from the head of state in Taiwan (especially when the DPP is in power). As the NYT detailed in a roundtable of experts, Trump's phone call has serious repercussions for keeping the status quo in the region.
WHY THIS MATTERS:
Trump used China as a punching bag during the campaign, saying they're stealing our jobs and tanking our economy. But as Obama said in his last press conference, the Chinese respond very differently when it comes to the subject of Taiwan. Trump will need to tread carefully.